Icosahedral symmetry

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</button> <ul id="toc-As_point_group-sublist" class="vector-toc-list"> <li id="toc-Visualizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visualizations"> <div class="vector-toc-text"> 2.1 Visualizations </div> </a> <ul id="toc-Visualizations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Group_structure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Group_structure"> <div class="vector-toc-text"> 3 Group structure </div> </a> <button aria-controls="toc-Group_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Group structure subsection </button> <ul id="toc-Group_structure-sublist" class="vector-toc-list"> <li id="toc-Isomorphism_of_I_with_A5" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isomorphism_of_I_with_A5"> <div class="vector-toc-text"> 3.1 Isomorphism of I with A5 </div> </a> <ul id="toc-Isomorphism_of_I_with_A5-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Commonly_confused_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Commonly_confused_groups"> <div class="vector-toc-text"> 3.2 Commonly confused groups </div> </a> <ul id="toc-Commonly_confused_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjugacy_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conjugacy_classes"> <div class="vector-toc-text"> 3.3 Conjugacy classes </div> </a> <ul id="toc-Conjugacy_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgroups_of_the_full_icosahedral_symmetry_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subgroups_of_the_full_icosahedral_symmetry_group"> <div class="vector-toc-text"> 3.4 Subgroups of the full icosahedral symmetry group </div> </a> <ul id="toc-Subgroups_of_the_full_icosahedral_symmetry_group-sublist" class="vector-toc-list"> <li id="toc-Vertex_stabilizers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vertex_stabilizers"> <div class="vector-toc-text"> 3.4.1 Vertex stabilizers </div> </a> <ul id="toc-Vertex_stabilizers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Edge_stabilizers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Edge_stabilizers"> <div class="vector-toc-text"> 3.4.2 Edge stabilizers </div> </a> <ul id="toc-Edge_stabilizers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Face_stabilizers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Face_stabilizers"> <div class="vector-toc-text"> 3.4.3 Face stabilizers </div> </a> <ul id="toc-Face_stabilizers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polyhedron_stabilizers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polyhedron_stabilizers"> <div class="vector-toc-text"> 3.4.4 Polyhedron stabilizers </div> </a> <ul id="toc-Polyhedron_stabilizers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coxeter_group_generators" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Coxeter_group_generators"> <div class="vector-toc-text"> 3.4.5 Coxeter group generators </div> </a> <ul id="toc-Coxeter_group_generators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Fundamental_domain" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fundamental_domain"> <div class="vector-toc-text"> 4 Fundamental domain </div> </a> <ul id="toc-Fundamental_domain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polyhedra_with_icosahedral_symmetry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polyhedra_with_icosahedral_symmetry"> <div class="vector-toc-text"> 5 Polyhedra with icosahedral symmetry </div> </a> <button aria-controls="toc-Polyhedra_with_icosahedral_symmetry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Polyhedra with icosahedral symmetry subsection </button> <ul id="toc-Polyhedra_with_icosahedral_symmetry-sublist" class="vector-toc-list"> <li id="toc-Chiral_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chiral_polyhedra"> <div class="vector-toc-text"> 5.1 Chiral polyhedra </div> </a> <ul id="toc-Chiral_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Full_icosahedral_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Full_icosahedral_symmetry"> <div class="vector-toc-text"> 5.2 Full icosahedral symmetry </div> </a> <ul id="toc-Full_icosahedral_symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_objects_with_icosahedral_symmetry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_objects_with_icosahedral_symmetry"> <div class="vector-toc-text"> 6 Other objects with icosahedral symmetry </div> </a> <button aria-controls="toc-Other_objects_with_icosahedral_symmetry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other objects with icosahedral symmetry subsection </button> <ul id="toc-Other_objects_with_icosahedral_symmetry-sublist" class="vector-toc-list"> <li id="toc-Liquid_crystals_with_icosahedral_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liquid_crystals_with_icosahedral_symmetry"> <div class="vector-toc-text"> 6.1 Liquid crystals with icosahedral symmetry </div> </a> <ul id="toc-Liquid_crystals_with_icosahedral_symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> 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(May 2020) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <table class="wikitable" style="float:right; margin-left:1em" width="300"> <caption>Selected <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> </caption> <tbody><tr style="text-align:center;"> <td><a href="/wiki/File:Sphere_symmetry_group_cs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/100px-Sphere_symmetry_group_cs.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/150px-Sphere_symmetry_group_cs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/200px-Sphere_symmetry_group_cs.png 2x" data-file-width="636" data-file-height="622" /></a> <a href="/wiki/List_of_spherical_symmetry_groups#Involutional_symmetry" title="List of spherical symmetry groups">Involutional symmetry</a> Cs, (*) [ ] = <img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><a href="/wiki/File:Sphere_symmetry_group_c3v.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/100px-Sphere_symmetry_group_c3v.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/150px-Sphere_symmetry_group_c3v.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/200px-Sphere_symmetry_group_c3v.png 2x" data-file-width="621" data-file-height="620" /></a> <a href="/wiki/Cyclic_symmetry_in_three_dimensions" title="Cyclic symmetry in three dimensions">Cyclic symmetry</a> Cnv, (*nn) [n] = <img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/4/46/CDel_n.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><a href="/wiki/File:Sphere_symmetry_group_d3h.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/100px-Sphere_symmetry_group_d3h.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/150px-Sphere_symmetry_group_d3h.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/200px-Sphere_symmetry_group_d3h.png 2x" data-file-width="621" data-file-height="620" /></a> <a href="/wiki/Dihedral_symmetry_in_three_dimensions" title="Dihedral symmetry in three dimensions">Dihedral symmetry</a> Dnh, (*n22) [n,2] = <img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/4/46/CDel_n.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td></tr> <tr> <th colspan="4"><a href="/wiki/Polyhedral_group" title="Polyhedral group">Polyhedral group</a>, [n,3], (*n32) </th></tr> <tr style="text-align:center;"> <td><a href="/wiki/File:Sphere_symmetry_group_td.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/100px-Sphere_symmetry_group_td.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/150px-Sphere_symmetry_group_td.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/200px-Sphere_symmetry_group_td.png 2x" data-file-width="649" data-file-height="625" /></a> <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">Tetrahedral symmetry</a> Td, (*332) [3,3] = <img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><a href="/wiki/File:Sphere_symmetry_group_oh.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/100px-Sphere_symmetry_group_oh.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/150px-Sphere_symmetry_group_oh.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/200px-Sphere_symmetry_group_oh.png 2x" data-file-width="649" data-file-height="625" /></a> <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">Octahedral symmetry</a> Oh, (*432) [4,3] = <img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><a href="/wiki/File:Sphere_symmetry_group_ih.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/100px-Sphere_symmetry_group_ih.png" decoding="async" width="100" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/150px-Sphere_symmetry_group_ih.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/200px-Sphere_symmetry_group_ih.png 2x" data-file-width="671" data-file-height="617" /></a> <a class="mw-selflink selflink">Icosahedral symmetry</a> Ih, (*532) [5,3] = <img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedral_reflection_domains.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/220px-Icosahedral_reflection_domains.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/330px-Icosahedral_reflection_domains.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedral_reflection_domains.png/440px-Icosahedral_reflection_domains.png 2x" data-file-width="811" data-file-height="812" /></a><figcaption>Icosahedral symmetry fundamental domains</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Soccer_ball.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/e/ec/Soccer_ball.svg/220px-Soccer_ball.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/ec/Soccer_ball.svg/330px-Soccer_ball.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/ec/Soccer_ball.svg/440px-Soccer_ball.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>A <a href="/wiki/Ball_(association_football)" title="Ball (association football)">soccer ball</a>, a common example of a <a href="/wiki/Spherical_polyhedron" title="Spherical polyhedron">spherical</a> <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a>, has full icosahedral symmetry.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sixteenth_stellation_of_icosahedron.png" class="mw-file-description"><img alt="A great icosahedron" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/200px-Sixteenth_stellation_of_icosahedron.png" decoding="async" width="200" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/300px-Sixteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/400px-Sixteenth_stellation_of_icosahedron.png 2x" data-file-width="940" data-file-height="900" /></a><figcaption>Rotations and reflections form the symmetry group of a <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a>.</figcaption></figure> In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> as a <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>. Examples of other <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> with icosahedral symmetry include the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> (the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> of the icosahedron) and the <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a>. Every polyhedron with icosahedral symmetry has 60 <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational</a> (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection</a>), for a total <a href="/wiki/Symmetry_order" class="mw-redirect" title="Symmetry order">symmetry order</a> of 120. The full <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> is the <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> of type H3. It may be represented by <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> [5,3] and <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagram</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" />. The set of rotational symmetries forms a subgroup that is isomorphic to the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> A5 on 5 letters. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=1" title="Edit section: Description">edit</a>]</div> Icosahedral symmetry is a mathematical property of objects indicating that an object has the same <a href="/wiki/Symmetry" title="Symmetry">symmetries</a> as a <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>. <div class="mw-heading mw-heading2"><h2 id="As_point_group">As point group</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=2" title="Edit section: As point group">edit</a>]</div> Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">discrete point symmetries</a> (or equivalently, <a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">symmetries on the sphere</a>) with the largest <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a>. Icosahedral symmetry is not compatible with <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>, so there are no associated <a href="/wiki/Crystal_system#Overview_of_point_groups_by_crystal_system" title="Crystal system">crystallographic point groups</a> or <a href="/wiki/Space_group" title="Space group">space groups</a>. <table class="wikitable"> <tbody><tr> <th><a href="/wiki/Arthur_Moritz_Sch%C3%B6nflies" class="mw-redirect" title="Arthur Moritz Schönflies">Schö.</a> </th> <th colspan="2"><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter</a> </th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a> </th> <th>Abstract structure </th> <th><a href="/wiki/Symmetry_order" class="mw-redirect" title="Symmetry order">Order</a> </th></tr> <tr align="center"> <td>I</td> <td>[5,3]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>532</td> <td><a href="/wiki/Alternating_group" title="Alternating group">A5</a></td> <td>60 </td></tr> <tr align="center"> <td>Ih</td> <td>[5,3]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*532</td> <td>A5×2</td> <td>120 </td></tr></tbody></table> <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">Presentations</a> corresponding to the above are: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∣</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2451a3262390b371bc95c9cb1f9626e54b465788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.3ex; height:3.176ex;" alt="{\displaystyle I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle \ }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle .\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∣</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle .\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5386a338b501fad22176c7b8db3bc1b7262c623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.681ex; height:3.176ex;" alt="{\displaystyle I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle .\ }"></dd></dl> These correspond to the icosahedral groups (rotational and full) being the (2,3,5) <a href="/wiki/Triangle_group" title="Triangle group">triangle groups</a>. The first presentation was given by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> in 1856, in his paper on <a href="/wiki/Icosian_calculus" title="Icosian calculus">icosian calculus</a>.<a href="#cite_note-1">[1]</a> Note that other presentations are possible, for instance as an <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> (for I). <div class="mw-heading mw-heading3"><h3 id="Visualizations">Visualizations</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=3" title="Edit section: Visualizations">edit</a>]</div> The full <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> is the <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> of type H3. It may be represented by <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> [5,3] and <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagram</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" />. The set of rotational symmetries forms a subgroup that is isomorphic to the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> A5 on 5 letters. <table class="wikitable"> <tbody><tr valign="top"> <th rowspan="2"><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoe.</a> (<a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a>) </th> <th rowspan="2"><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> </th> <th rowspan="2">Elements </th> <th colspan="4">Mirror diagrams </th></tr> <tr> <th>Orthogonal </th> <th colspan="3"><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> </th></tr> <tr align="center"> <th>Ih (*532) </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> <img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> [5,3]</td> <td>Mirror lines: 15 <img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><a href="/wiki/File:Spherical_disdyakis_triacontahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Spherical_disdyakis_triacontahedron.png/120px-Spherical_disdyakis_triacontahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Spherical_disdyakis_triacontahedron.png/180px-Spherical_disdyakis_triacontahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Spherical_disdyakis_triacontahedron.png/240px-Spherical_disdyakis_triacontahedron.png 2x" data-file-width="564" data-file-height="562" /></a> </td> <td><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Disdyakis_triacontahedron_stereographic_d5.svg/150px-Disdyakis_triacontahedron_stereographic_d5.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Disdyakis_triacontahedron_stereographic_d5.svg/225px-Disdyakis_triacontahedron_stereographic_d5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Disdyakis_triacontahedron_stereographic_d5.svg/300px-Disdyakis_triacontahedron_stereographic_d5.svg.png 2x" data-file-width="960" data-file-height="960" /></a> </td> <td><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Disdyakis_triacontahedron_stereographic_d3.svg/150px-Disdyakis_triacontahedron_stereographic_d3.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Disdyakis_triacontahedron_stereographic_d3.svg/225px-Disdyakis_triacontahedron_stereographic_d3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Disdyakis_triacontahedron_stereographic_d3.svg/300px-Disdyakis_triacontahedron_stereographic_d3.svg.png 2x" data-file-width="960" data-file-height="960" /></a> </td> <td><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Disdyakis_triacontahedron_stereographic_d2.svg/150px-Disdyakis_triacontahedron_stereographic_d2.svg.png" decoding="async" width="150" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Disdyakis_triacontahedron_stereographic_d2.svg/225px-Disdyakis_triacontahedron_stereographic_d2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Disdyakis_triacontahedron_stereographic_d2.svg/300px-Disdyakis_triacontahedron_stereographic_d2.svg.png 2x" data-file-width="1200" data-file-height="1000" /></a> </td></tr> <tr align="center"> <th>I (532) </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> <a href="/wiki/File:Coxeter_diagram_chiral_icosahedral_group.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e0/Coxeter_diagram_chiral_icosahedral_group.png" decoding="async" width="30" height="31" class="mw-file-element" data-file-width="30" data-file-height="31" /></a> [5,3]+</td> <td>Gyration points: 125<a href="/wiki/File:Patka_piechota.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/12px-Patka_piechota.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/18px-Patka_piechota.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/24px-Patka_piechota.png 2x" data-file-width="1030" data-file-height="1004" /></a> 203<a href="/wiki/File:Armed_forces_red_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/12px-Armed_forces_red_triangle.svg.png" decoding="async" width="12" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/18px-Armed_forces_red_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/24px-Armed_forces_red_triangle.svg.png 2x" data-file-width="270" data-file-height="240" /></a> 302<a href="/wiki/File:Rhomb.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/12px-Rhomb.svg.png" decoding="async" width="12" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/18px-Rhomb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/24px-Rhomb.svg.png 2x" data-file-width="63" data-file-height="80" /></a> </td> <td><a href="/wiki/File:Sphere_symmetry_group_i.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/120px-Sphere_symmetry_group_i.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/180px-Sphere_symmetry_group_i.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/240px-Sphere_symmetry_group_i.png 2x" data-file-width="985" data-file-height="987" /></a> </td> <td valign="bottom"><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d5_gyrations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Disdyakis_triacontahedron_stereographic_d5_gyrations.png/150px-Disdyakis_triacontahedron_stereographic_d5_gyrations.png" decoding="async" width="150" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Disdyakis_triacontahedron_stereographic_d5_gyrations.png/225px-Disdyakis_triacontahedron_stereographic_d5_gyrations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Disdyakis_triacontahedron_stereographic_d5_gyrations.png/300px-Disdyakis_triacontahedron_stereographic_d5_gyrations.png 2x" data-file-width="665" data-file-height="589" /></a> <a href="/wiki/File:Patka_piechota.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/12px-Patka_piechota.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/18px-Patka_piechota.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Patka_piechota.png/24px-Patka_piechota.png 2x" data-file-width="1030" data-file-height="1004" /></a> </td> <td valign="bottom"><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d3_gyrations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Disdyakis_triacontahedron_stereographic_d3_gyrations.png/150px-Disdyakis_triacontahedron_stereographic_d3_gyrations.png" decoding="async" width="150" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Disdyakis_triacontahedron_stereographic_d3_gyrations.png/225px-Disdyakis_triacontahedron_stereographic_d3_gyrations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Disdyakis_triacontahedron_stereographic_d3_gyrations.png/300px-Disdyakis_triacontahedron_stereographic_d3_gyrations.png 2x" data-file-width="827" data-file-height="588" /></a> <a href="/wiki/File:Armed_forces_red_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/12px-Armed_forces_red_triangle.svg.png" decoding="async" width="12" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/18px-Armed_forces_red_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Armed_forces_red_triangle.svg/24px-Armed_forces_red_triangle.svg.png 2x" data-file-width="270" data-file-height="240" /></a> </td> <td valign="bottom"><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d2_gyrations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Disdyakis_triacontahedron_stereographic_d2_gyrations.png/150px-Disdyakis_triacontahedron_stereographic_d2_gyrations.png" decoding="async" width="150" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Disdyakis_triacontahedron_stereographic_d2_gyrations.png/225px-Disdyakis_triacontahedron_stereographic_d2_gyrations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Disdyakis_triacontahedron_stereographic_d2_gyrations.png/300px-Disdyakis_triacontahedron_stereographic_d2_gyrations.png 2x" data-file-width="945" data-file-height="589" /></a> <a href="/wiki/File:Rhomb.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/12px-Rhomb.svg.png" decoding="async" width="12" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/18px-Rhomb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Rhomb.svg/24px-Rhomb.svg.png 2x" data-file-width="63" data-file-height="80" /></a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Group_structure">Group structure</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=4" title="Edit section: Group structure">edit</a>]</div> Every <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> with icosahedral symmetry has 60 <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational</a> (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection</a>), for a total <a href="/wiki/Symmetry_order" class="mw-redirect" title="Symmetry order">symmetry order</a> of 120. <table class="wikitable" width="320" align="right"> <tbody><tr> <td><a href="/wiki/File:Spherical_compound_of_five_octahedra.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Spherical_compound_of_five_octahedra.png/160px-Spherical_compound_of_five_octahedra.png" decoding="async" width="160" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Spherical_compound_of_five_octahedra.png/240px-Spherical_compound_of_five_octahedra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Spherical_compound_of_five_octahedra.png/320px-Spherical_compound_of_five_octahedra.png 2x" data-file-width="603" data-file-height="612" /></a> </td> <td><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d2_5-color.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Disdyakis_triacontahedron_stereographic_d2_5-color.png/200px-Disdyakis_triacontahedron_stereographic_d2_5-color.png" decoding="async" width="200" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Disdyakis_triacontahedron_stereographic_d2_5-color.png/300px-Disdyakis_triacontahedron_stereographic_d2_5-color.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Disdyakis_triacontahedron_stereographic_d2_5-color.png/400px-Disdyakis_triacontahedron_stereographic_d2_5-color.png 2x" data-file-width="934" data-file-height="592" /></a> </td></tr> <tr> <td colspan="2">The edges of a spherical <a href="/wiki/Compound_of_five_octahedra" title="Compound of five octahedra">compound of five octahedra</a> represent the 15 mirror planes as colored great circles. Each octahedron can represent 3 orthogonal mirror planes by its edges. </td></tr> <tr> <td><a href="/wiki/File:Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png/160px-Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png" decoding="async" width="160" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png/240px-Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png/320px-Spherical_compound_of_five_octahedra-pyritohedral_symmetry.png 2x" data-file-width="602" data-file-height="612" /></a> </td> <td><a href="/wiki/File:Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png/200px-Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png" decoding="async" width="200" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png/300px-Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png/400px-Disdyakis_triacontahedron_stereographic_d2_pyritohedral.png 2x" data-file-width="934" data-file-height="592" /></a> </td></tr> <tr> <td colspan="2">The <a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">pyritohedral symmetry</a> is an index 5 subgroup of icosahedral symmetry, with 3 orthogonal green reflection lines and 8 red order-3 gyration points. There are 5 different orientations of pyritohedral symmetry. </td></tr></tbody></table> The <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>icosahedral rotation group I is of order 60. The group I is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to A5, the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the <a href="/wiki/Compound_of_five_cubes" title="Compound of five cubes">compound of five cubes</a> (which inscribe in the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>), the <a href="/wiki/Compound_of_five_octahedra" title="Compound of five octahedra">compound of five octahedra</a>, or either of the two <a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">compounds of five tetrahedra</a> (which are <a href="/wiki/Enantiomorphs" class="mw-redirect" title="Enantiomorphs">enantiomorphs</a>, and inscribe in the dodecahedron). The group contains 5 versions of Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5. The <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">full icosahedral group Ih has order 120. It has I as <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the <a href="/wiki/Inversion_in_a_point" class="mw-redirect" title="Inversion in a point">inversion in the center</a> corresponding to element (identity,-1), where Z2 is written multiplicatively. Ih acts on the <a href="/wiki/Compound_of_five_cubes" title="Compound of five cubes">compound of five cubes</a> and the <a href="/wiki/Compound_of_five_octahedra" title="Compound of five octahedra">compound of five octahedra</a>, but −1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the <a href="/wiki/Compound_of_ten_tetrahedra" title="Compound of ten tetrahedra">compound of ten tetrahedra</a>: I acts on the two chiral halves (<a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">compounds of five tetrahedra</a>), and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic; see below for details. The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms). I is also isomorphic to PSL2(5), but Ih is not isomorphic to SL2(5). <div class="mw-heading mw-heading3"><h3 id="Isomorphism_of_I_with_A5">Isomorphism of I with A5</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=5" title="Edit section: Isomorphism of I with A5">edit</a>]</div> It is useful to describe explicitly what the isomorphism between I and A5 looks like. In the following table, permutations Pi and Qi act on 5 and 12 elements respectively, while the rotation matrices Mi are the elements of I. If Pk is the product of taking the permutation Pi and applying Pj to it, then for the same values of i, j and k, it is also true that Qk is the product of taking Qi and applying Qj, and also that premultiplying a vector by Mk is the same as premultiplying that vector by Mi and then premultiplying that result with Mj, that is Mk = Mj × Mi. Since the permutations Pi are all the 60 even permutations of 12345, the <a href="/wiki/Bijection" title="Bijection">one-to-one correspondence</a> is made explicit, therefore the isomorphism too. <table class="wikitable collapsible collapsed" align="center" style="font-family:'DejaVu Sans Mono','monospace'"> <tbody><tr> <th width="25%">Rotation matrix </th> <th width="25%">Permutation of 5 on 1 2 3 4 5 </th> <th width="50%">Permutation of 12 on 1 2 3 4 5 6 7 8 9 10 11 12 </th></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a806a740b2eade1b3624aa8aae7bba44116fc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.391ex; height:9.176ex;" alt="{\displaystyle M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"> = () </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ea6463cb36d8278ff71214fb4d13127039ae53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}"> = () </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53856a09282da581d12f64c1cc792edc98e7996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.262ex; height:13.843ex;" alt="{\displaystyle M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"> = (3 4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86e8bff64d5e62fc8f45a35875e78bc9bef74a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{2}}"> = (1 11 8)(2 9 6)(3 5 12)(4 7 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{3}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{3}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869753f9815790a8f340814af314cc8f023aaffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.262ex; height:13.843ex;" alt="{\displaystyle M_{3}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"> = (3 5 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b6c2a91a49263a333768fc2ebebdc379ddf5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{3}}"> = (1 8 11)(2 6 9)(3 12 5)(4 10 7) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{4}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{4}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb1cee779ca8c351ed09634de607018ce05d019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.776ex; height:13.843ex;" alt="{\displaystyle M_{4}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"> = (2 3)(4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c450d0ca92013fa4e40c20c47768466d0a71ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{4}}"> = (1 12)(2 8)(3 6)(4 9)(5 10)(7 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{5}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{5}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9cb6269c5891b7e981edafddf1c13f88b55e3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.42ex; height:13.843ex;" alt="{\displaystyle M_{5}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"> = (2 3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd88f6ad0c94313d98590e698dc5c2868e94071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{5}}"> = (1 2 3)(4 5 6)(7 9 8)(10 11 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{6}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{6}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd8949fcd75a808d37c13f389c07f0b66ae3a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.42ex; height:13.843ex;" alt="{\displaystyle M_{6}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"> = (2 3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e42a16164b1e0be76312f5fe177c9bb7b41937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{6}}"> = (1 7 5)(2 4 11)(3 10 9)(6 8 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{7}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{7}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2ff1aec9bdf60293b51f105f0494d2c67771e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:26.434ex; height:13.843ex;" alt="{\displaystyle M_{7}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{7}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada04c60312be4ff31aa8131f3aefe12e285449d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{7}}"> = (2 4 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{7}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2745b4dcfb278ef072efb99935b211149039d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{7}}"> = (1 3 2)(4 6 5)(7 8 9)(10 12 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{8}={\begin{bmatrix}0&-1&0\\0&0&1\\-1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{8}={\begin{bmatrix}0&-1&0\\0&0&1\\-1&0&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ffc1e91a07ad14bd47757f41c2592cc0679e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.008ex; height:9.176ex;" alt="{\displaystyle M_{8}={\begin{bmatrix}0&-1&0\\0&0&1\\-1&0&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{8}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dc39edfd481c14b9bec617ba49f675b9d5107b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{8}}"> = (2 4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{8}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e87378c5c1811891cbc1e5c334b2c1ba4b84d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{8}}"> = (1 10 6)(2 7 12)(3 4 8)(5 11 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{9}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{9}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cb210dbb2127b5337eadcfc490f9d77deebd0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:24.632ex; height:13.843ex;" alt="{\displaystyle M_{9}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{9}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb74c250e77f086cccd7724a80c6210e4344bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{9}}"> = (2 4)(3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{9}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6efcf0d263472c48649110e18268619efa77317d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{9}}"> = (1 9)(2 5)(3 11)(4 12)(6 7)(8 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{10}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{10}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09cf863efbe97e73b3ecc1fccd1ab63872c1bd66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{10}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{10}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ba3dedab8f977111ccb2fb6495105684ae007a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{10}}"> = (2 5 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{10}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3c9c06a9e2a32b3b395bc7632593c146f7d412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{10}}"> = (1 5 7)(2 11 4)(3 9 10)(6 12 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{11}={\begin{bmatrix}0&0&-1\\-1&0&0\\0&1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{11}={\begin{bmatrix}0&0&-1\\-1&0&0\\0&1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ff8110ffc8feb96347137fad8028bf4cef6bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{11}={\begin{bmatrix}0&0&-1\\-1&0&0\\0&1&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{11}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ed365f0389ecd7c5159589246f8962bcab211f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{11}}"> = (2 5 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{11}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4a436d88f2b0d4d35ac935ac179657786828b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{11}}"> = (1 6 10)(2 12 7)(3 8 4)(5 9 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{12}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{12}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15231d19c20e541f5b9551983ecd373173f8e2d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{12}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7004712437b915c081c9f53451c9cadce4061863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{12}}"> = (2 5)(3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f8f748898a063e66d70ac80acb023099ef82d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{12}}"> = (1 4)(2 10)(3 7)(5 8)(6 11)(9 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{13}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{13}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff653e3ff530f72d9f8244f04fcf5d51b686dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{13}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{13}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9310fd1f3a3589444006baa707f72739761d3356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{13}}"> = (1 2)(4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{13}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578b199baec4a626ff027726610a0759582a252d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{13}}"> = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{14}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{14}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362c718ce98b647cd63fe24ce3112e109404a098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{14}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{14}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{14}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c647d9970b8e27302b4a448f2fe6d2bec0e1c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{14}}"> = (1 2)(3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{14}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{14}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379614ca2f05fcbdb131a138d8fe8490633e81ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{14}}"> = (1 5)(2 7)(3 11)(4 9)(6 10)(8 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{15}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{15}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99815f70f1b2ca01a2c86d05e76c07192ec18dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{15}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{15}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06f0dc3713cd9f803b00f9efdb1085369b6b269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{15}}"> = (1 2)(3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{15}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9deba7e9f00e9edb96eb76c535fdb9f2a52a585c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{15}}"> = (1 12)(2 10)(3 8)(4 6)(5 11)(7 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{16}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{16}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ce9f95b3a4702262fcda2062acf3570c7d3228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{16}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{16}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{16}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42c8029bd6f5641034e378f353ab2f62b074eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{16}}"> = (1 2 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{16}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{16}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba93838d0896e645d6deb7dee3ed433a35d03b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{16}}"> = (1 11 6)(2 5 9)(3 7 12)(4 10 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{17}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{17}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cce4e5f672f20bb0438fe2c81af55585cdc6180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{17}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{17}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{17}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c29941ff0a902b0f9d9e19b25573086241a84a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{17}}"> = (1 2 3 4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{17}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{17}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7060b35925f54db26dbaea8126c1b48bac6150d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{17}}"> = (1 6 5 3 9)(4 12 7 8 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{18}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{18}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d08bcde304d68d838534ccf4f7e0b4f4a72985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{18}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{18}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{18}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9c2c21334a2300d8c0468f14f2942fea086d3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{18}}"> = (1 2 3 5 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{18}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{18}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6386dd33c48c411d9213e01434f67ffed783faa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{18}}"> = (1 4 8 6 2)(5 7 10 12 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{19}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{19}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba06542618b7919bc610fcfed839a1285904fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{19}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{19}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{19}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982cf32b421807b0511466bcc924496606546b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{19}}"> = (1 2 4 5 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{19}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{19}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec129dcce39948e034a67f5a7eb5792e31a56bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{19}}"> = (1 8 7 3 10)(2 12 5 6 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{20}={\begin{bmatrix}0&0&1\\-1&0&0\\0&-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{20}={\begin{bmatrix}0&0&1\\-1&0&0\\0&-1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9fad3d4cb99a6b3d7063603df9c07ddf57aaf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{20}={\begin{bmatrix}0&0&1\\-1&0&0\\0&-1&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{20}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{20}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ffc819301cf072f7fa6ccc3078e79a15e02bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{20}}"> = (1 2 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{20}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{20}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4785513a97795e1e4cca78ac396d21ddcae89d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{20}}"> = (1 7 4)(2 11 8)(3 5 10)(6 9 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{21}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{21}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed7f9e6c53355f01645360455ea5c0623a05599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{21}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{21}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8d11e3ae15dde0a77bc9c9863bbbf5694d0b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{21}}"> = (1 2 4 3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{21}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2225037cd6255b3c4a4c23716cbfaa5730892f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{21}}"> = (1 2 9 11 7)(3 6 12 10 4) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{22}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{22}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4a4dc00e27d076b8325b31492f892fd0e7093e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{22}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{22}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{22}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ee776028f8e9ce699fac5d53fad4173ba04f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{22}}"> = (1 2 5 4 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{22}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{22}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a690383c0bb27a954c9cdef88a23618fc6703a3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{22}}"> = (2 3 4 7 5)(6 8 10 11 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{23}={\begin{bmatrix}0&1&0\\0&0&-1\\-1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{23}={\begin{bmatrix}0&1&0\\0&0&-1\\-1&0&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d165834f35a29a7112ab7ef70e10aa68b4648cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{23}={\begin{bmatrix}0&1&0\\0&0&-1\\-1&0&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{23}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df367250134b7eba998805de9bae274e813ad94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{23}}"> = (1 2 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{23}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db8d2149383fe3b4609bb4136e0b3638ffebf05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{23}}"> = (1 9 8)(2 6 3)(4 5 12)(7 11 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{24}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{24}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c4dffd81d6574f0dfd098967f66ba60201ce40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{24}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{24}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{24}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247559482bc9939502e11f991b16169de4a1834c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{24}}"> = (1 2 5 3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{24}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{24}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0e0e465460d57707ab2ee7d9b3e01243a2de16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{24}}"> = (1 10 5 4 11)(2 8 9 3 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{25}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{25}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b7e76e993c075fe2a9f741ec72f3d3b27eb5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{25}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{25}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{25}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f3b6b738f3b9214a57dbd9907cd51473e76ba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{25}}"> = (1 3 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{25}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{25}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6397d13f4d7b30332eafe37b672b396a023065d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{25}}"> = (1 6 11)(2 9 5)(3 12 7)(4 8 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{26}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{26}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977b11d5553fcc0eb973dd3a0aa974dcbb3c1f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{26}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{26}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{26}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46236165cff046069a89b6731e5c46459491ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{26}}"> = (1 3 4 5 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{26}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{26}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a429d8ad379b7d585f64ce58203682820ac29a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{26}}"> = (2 5 7 4 3)(6 9 11 10 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{27}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{27}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5c53129066c378a0fac1b96ecc3236fcdd79ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{27}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{27}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{27}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9073aec51254c136ff1c6d53462700ee9f490510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{27}}"> = (1 3 5 4 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{27}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{27}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b211eea7f170597baae1c0fabb2e18e8d36a7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{27}}"> = (1 10 3 7 8)(2 11 6 5 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{28}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>28</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{28}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9845991751f096b9b8dbaaf84d20c6685b054cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{28}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{28}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>28</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{28}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f781cb85b488874ad9e20200d03cd78b043075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{28}}"> = (1 3)(4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{28}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>28</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{28}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea1adc89e5d63029f378c469ac681117fcca5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{28}}"> = (1 7)(2 10)(3 11)(4 5)(6 12)(8 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{29}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>29</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{29}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc40f0e347366313bcd25e3cbef61c82971bb32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{29}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{29}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>29</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{29}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9828d2f7a9a3dd2bb6b7250951722c741de0ba51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{29}}"> = (1 3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{29}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>29</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{29}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3bdf8bc893948d32e4655c40edb0c3dc489068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{29}}"> = (1 9 10)(2 12 4)(3 6 8)(5 11 7) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{30}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{30}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111e67ce276143b346ffcfd0276c0894753f0258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{30}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{30}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{30}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ab472253d9c7914eaf0582bac99e34a2f2d08ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{30}}"> = (1 3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{30}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{30}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da8f6452145f06adb92019fca6438d1ded8ddd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{30}}"> = (1 3 4)(2 8 7)(5 6 10)(9 12 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{31}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{31}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a56beb25402b0f07d35826901a3ba01877e165b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{31}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{31}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{31}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49e5ecdb4ae7c281b09007ba5ba6cef523bb5b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{31}}"> = (1 3)(2 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{31}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{31}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc781dc65cdc301887ad324a04eb9e5da28376c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{31}}"> = (1 12)(2 6)(3 9)(4 11)(5 8)(7 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{32}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{32}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5e7cd058eb9685208e07a550dc277211118712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{32}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{32}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{32}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d11709eccc6da5af8b7edf3098a9682ea49c456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{32}}"> = (1 3 2 4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{32}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{32}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825b1fca218d27ff43b9ab8b94802a75f72bddcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{32}}"> = (1 4 10 11 5)(2 3 8 12 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{33}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{33}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5521942b0b8628499374ffb3d302c366f2f1d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{33}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{33}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{33}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287b7d4bd9b562833d7939e758845bfb9c1fe719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{33}}"> = (1 3 5 2 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{33}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{33}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a95d6aa3cdea16f6c5b755f7d3ebaae4b8104a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{33}}"> = (1 5 9 6 3)(4 7 11 12 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{34}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{34}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f41db12ff64ab871ff50f97f3d391167d1208ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{34}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{34}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{34}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c6ceef0fb3e15d95ea46ef867038391207b318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{34}}"> = (1 3)(2 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{34}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{34}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4878158ae6ba5a1abb38c8b3559fe400b9d5497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{34}}"> = (1 2)(3 5)(4 9)(6 7)(8 11)(10 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{35}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{35}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c154dc633c6d7aefa0b54bb6444ecf6ab546b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{35}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{35}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{35}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8e0c426047c6139e036b6a23e20e681909b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{35}}"> = (1 3 2 5 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{35}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{35}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726b558e171f84d8af86630a0c1f3be5058bff25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{35}}"> = (1 11 2 7 9)(3 10 6 4 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{36}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{36}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68f0878e52fed098b0fd9c1d5663b38cc69b53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{36}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{36}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{36}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404c56ef7a23cbf7f2e341308ead634951cfc781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{36}}"> = (1 3 4 2 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{36}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{36}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcdb54235724f53f2161422451f44567c1daca36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{36}}"> = (1 8 2 4 6)(5 10 9 7 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{37}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>37</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{37}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917441bbc4f43bbb175f753f0bf81ce38034f719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{37}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{37}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>37</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{37}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8691b7f462c454b3ab35e8093c5b8ab6cec73148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{37}}"> = (1 4 5 3 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{37}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>37</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{37}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3448453c0526dd73532d606270d70b6a9e53f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{37}}"> = (1 2 6 8 4)(5 9 12 10 7) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{38}={\begin{bmatrix}0&-1&0\\0&0&-1\\1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>38</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{38}={\begin{bmatrix}0&-1&0\\0&0&-1\\1&0&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fa2e4d30186e6e33169a45198b87fac4fc2a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{38}={\begin{bmatrix}0&-1&0\\0&0&-1\\1&0&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{38}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>38</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{38}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9851c8a63bcece8b9dd4aa854e493438522049ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{38}}"> = (1 4 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{38}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>38</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{38}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d1cbe5f072ac9f92d3ba56a76946f6a83c7582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{38}}"> = (1 4 7)(2 8 11)(3 10 5)(6 12 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{39}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>39</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{39}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c33e783e2b1d8bcb6906cf38a106c853fba219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{39}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{39}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>39</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{39}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8444f3d3782d67d0e52bc5f33db40a41bf8a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{39}}"> = (1 4 3 5 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{39}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>39</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{39}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b25aa2dd2e9908eee11d61ffceaca5e838a261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{39}}"> = (1 11 4 5 10)(2 12 3 9 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{40}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>40</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{40}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f074f1b11bc2b3515704ac13bdb84bad27a8001a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{40}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{40}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>40</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{40}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace6c0798993357ef2588e279ac7b507fe570cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{40}}"> = (1 4 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{40}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>40</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{40}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd965158a4a557e62d97d97aeeb6444ed9783b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{40}}"> = (1 10 9)(2 4 12)(3 8 6)(5 7 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{41}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{41}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcfd1c0dae81df1a6ba5f69539032efce06fc179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.213ex; height:9.176ex;" alt="{\displaystyle M_{41}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{41}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{41}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c05a7a24652cdba31e21c50d6da6a5a46e647e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{41}}"> = (1 4 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{41}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{41}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38496081e0b202093332e45745b6c80312ef3cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{41}}"> = (1 5 2)(3 7 9)(4 11 6)(8 10 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{42}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{42}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1df8e3f5b998c8203cf7bc3d3d04dfbbf1cd360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{42}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{42}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{42}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acc62c0e53adce99d50fdff39332b32572d7413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{42}}"> = (1 4)(3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{42}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{42}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945220dbe550020da68fed754c52bc1264153448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{42}}"> = (1 6)(2 3)(4 9)(5 8)(7 12)(10 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{43}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{43}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612d8c230552b6fdd074356c31d62ef9c86fd2ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{43}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{43}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{43}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c23b206afa4387cca55440d0afc18c1517b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{43}}"> = (1 4 5 2 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{43}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{43}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bdb16483f46f2cfdb5d507f2731ee3425cdaeca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{43}}"> = (1 9 7 2 11)(3 12 4 6 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{44}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{44}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fabaf7bc93e59f3b00b4b49127228bff4356e278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:26.598ex; height:13.843ex;" alt="{\displaystyle M_{44}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{44}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{44}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ef56899ab0bf21c7100e441df853e9055b6c45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{44}}"> = (1 4)(2 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{44}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{44}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3646bf324a36263b702fd149171e83c4819fe23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{44}}"> = (1 8)(2 10)(3 4)(5 12)(6 7)(9 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{45}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{45}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1711ecf0a9964b405be019a26fad213017f7ba4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{45}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{45}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{45}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122e6743824a8b285470cf525d560e772a3b8284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{45}}"> = (1 4 2 3 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{45}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{45}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fa22aac4b5e7927b25fdf0d4e40d39be6e410d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{45}}"> = (2 7 3 5 4)(6 11 8 9 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{46}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{46}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b49d8ed67256dfbcf9bd9bed70ad467a33cfb0c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{46}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{46}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{46}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180974f60490d2e20daa5e2601331ba8a4dd375d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{46}}"> = (1 4 2 5 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{46}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{46}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a8fea62d781141ef744b4eb989d46f830f40b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{46}}"> = (1 3 6 9 5)(4 8 12 11 7) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{47}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>47</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{47}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da06f844e24470163423c1da43d457e8787156b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{47}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{47}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>47</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{47}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a381565cbbab1bb5f659e5751ff2c2c41a59cbaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{47}}"> = (1 4 3 2 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{47}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>47</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{47}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878202545907def693005df1f85c78035f441a22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{47}}"> = (1 7 10 8 3)(2 5 11 12 6) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{48}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>48</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{48}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff76a0d971f8f71271b0d743f2ca1040a392f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{48}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{48}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>48</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{48}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d8a44bc7d4a44f268ea2d09d32de180c36debc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{48}}"> = (1 4)(2 5) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{48}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>48</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{48}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8147a5d1ce419b702eaa6f00c17c7c0eaf523cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{48}}"> = (1 12)(2 9)(3 11)(4 10)(5 6)(7 8) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{49}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>49</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{49}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb75b5e90e9e1e87abc7ae12956402ccab699f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{49}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{49}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>49</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{49}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6f9df4ed8fc6437c98294de3922bb52428c67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{49}}"> = (1 5 4 3 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{49}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>49</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{49}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7921702a3fbbc0e167e28c076a206b6ffbc944a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{49}}"> = (1 9 3 5 6)(4 11 8 7 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{50}={\begin{bmatrix}0&0&-1\\1&0&0\\0&-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{50}={\begin{bmatrix}0&0&-1\\1&0&0\\0&-1&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bcae77b283060f02342d19a9e8afd5f1ffc11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{50}={\begin{bmatrix}0&0&-1\\1&0&0\\0&-1&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{50}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{50}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ed8916510b4099d60a240f183258821f29d0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{50}}"> = (1 5 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{50}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{50}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3dde3f211b2ac081e04076c71dfa1b2850acca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{50}}"> = (1 8 9)(2 3 6)(4 12 5)(7 10 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{51}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>51</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{51}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c663a9a60b72f5c3428450946e1439482c64c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.454ex; height:13.843ex;" alt="{\displaystyle M_{51}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{51}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>51</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{51}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851991092c4e206bfddf702966554c952c5ee999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{51}}"> = (1 5 3 4 2) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{51}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>51</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{51}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53817e0ff55329a56f5fde4ce38d67fb9a4ffb11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{51}}"> = (1 7 11 9 2)(3 4 10 12 6) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{52}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>52</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{52}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e857dc884f78ddb38dde3fb1b7336a852d800e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{52}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{52}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>52</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{52}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89ecdc1ddb0970f864106cee44f049259612651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{52}}"> = (1 5 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{52}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>52</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{52}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1214e0ca0312104abcf2506e5c5bed6081bf7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{52}}"> = (1 4 3)(2 7 8)(5 10 6)(9 11 12) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{53}={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>53</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{53}={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b789540ae2850e145b94b5f5e8b4af46e1f0abe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.213ex; height:9.176ex;" alt="{\displaystyle M_{53}={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{53}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>53</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{53}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b62158eb9240063e5ba2048f19c2e176fd6d3354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{53}}"> = (1 5 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{53}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>53</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{53}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6944e644acfcf623500e906cbefffdc5b1981a91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{53}}"> = (1 2 5)(3 9 7)(4 6 11)(8 12 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{54}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>54</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{54}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2d600e08a9c3b84765f44bf6414048e30b8d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.084ex; height:13.843ex;" alt="{\displaystyle M_{54}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{54}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>54</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{54}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d526ad2d1c226a1ffb3d147fffebd7932154b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{54}}"> = (1 5)(3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{54}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>54</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{54}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9d47af72ce1bfbe11c353f0e2f1fef4d24f9b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{54}}"> = (1 12)(2 11)(3 10)(4 8)(5 9)(6 7) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{55}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{55}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ce9c6f2f30621d2a5208463dcd0c18d347377c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{55}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{55}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{55}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb544d64f92082b64019b292091111b35b2fe579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{55}}"> = (1 5 4 2 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{55}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{55}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0329473b8343df2f3430c211c6686c43ab90f361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{55}}"> = (1 5 11 10 4)(2 9 12 8 3) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{56}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{56}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557c5daa9766f6bbea5508b221325716a98862e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:26.598ex; height:13.843ex;" alt="{\displaystyle M_{56}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{56}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{56}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f7d0c4377a54393ccdb96e92867d27b507446d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{56}}"> = (1 5)(2 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{56}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{56}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0d4ad12125e600866557404691f14ce4c6b151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{56}}"> = (1 10)(2 12)(3 11)(4 7)(5 8)(6 9) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{57}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>57</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{57}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695e634c74b3a3725a5c406b4d8dd5ca69d6ece6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{57}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{57}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>57</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{57}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab906c60e68beefa89e178a464e71b956d7175e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{57}}"> = (1 5 2 3 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{57}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>57</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{57}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9fd60bb7a336e6fbe36b5d6f7ac014d8c2e511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{57}}"> = (1 3 8 10 7)(2 6 12 11 5) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{58}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>58</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{58}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a434c0809d51319ae1d77b4f90050cac0f6292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.242ex; height:13.843ex;" alt="{\displaystyle M_{58}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{58}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>58</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{58}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5382d4169c41cbc607f304f15179d809a48b49a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{58}}"> = (1 5 2 4 3) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{58}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>58</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{58}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9745821431e4e646b912e550e33ecd3bbb9b9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{58}}"> = (1 6 4 2 8)(5 12 7 9 10) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{59}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{59}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ad9219b7c5cecb41ce2f84269c75af5c869e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:27.256ex; height:13.843ex;" alt="{\displaystyle M_{59}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{59}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{59}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/050573b3bbc7f09cd2a394b3c841c99c66c6a81b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{59}}"> = (1 5 3 2 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{59}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{59}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4959d62a3c36572379c9d895145b48a5029ece9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{59}}"> = (2 4 5 3 7)(6 10 9 8 11) </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{60}={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>60</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{60}={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e798672e745d345cca86d598260e5e68571fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.83ex; height:9.176ex;" alt="{\displaystyle M_{60}={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{60}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>60</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{60}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29907080fd9822e7b482ce96ede16f288a69957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{60}}"> = (1 5)(2 4) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{60}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>60</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{60}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4715b9f338eaedf04acea0ccb928ae6010424aca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle Q_{60}}"> = (1 11)(2 10)(3 12)(4 9)(5 7)(6 8) </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Commonly_confused_groups">Commonly confused groups</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=6" title="Edit section: Commonly confused groups">edit</a>]</div> The following groups all have order 120, but are not isomorphic: <ul><li>S5, the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on 5 elements</li> <li>Ih, the full icosahedral group (subject of this article, also known as H3)</li> <li>2I, the <a href="/wiki/Binary_icosahedral_group" title="Binary icosahedral group">binary icosahedral group</a></li></ul> They correspond to the following <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequences</a> (the latter of which does not split) and product <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to A_{5}\to S_{5}\to Z_{2}\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to A_{5}\to S_{5}\to Z_{2}\to 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8035c227946848c9a772468095f9d37162b1e8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.699ex; height:2.509ex;" alt="{\displaystyle 1\to A_{5}\to S_{5}\to Z_{2}\to 1}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{h}=A_{5}\times Z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{h}=A_{5}\times Z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378423a36a631f97b40bb710c60f533651a66a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.58ex; height:2.509ex;" alt="{\displaystyle I_{h}=A_{5}\times Z_{2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to Z_{2}\to 2I\to A_{5}\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>2</mn> <mi>I</mi> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to Z_{2}\to 2I\to A_{5}\to 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d696170a4446731b645213e8636e9aed84437b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.554ex; height:2.509ex;" alt="{\displaystyle 1\to Z_{2}\to 2I\to A_{5}\to 1}"></dd></dl> In words, <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e213bbb69691c65e1391fe16cd79a0029471446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}"> is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d113353a42f71fc5e7154ddef2257079ab2e25a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{5}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e213bbb69691c65e1391fe16cd79a0029471446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}"> is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{h}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c84ca7860f66cd4ed8ecb07b4c5691f73c7365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle I_{h}}">, which is a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e213bbb69691c65e1391fe16cd79a0029471446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}"> is a <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dbd6170826fcd1a6f8c7b3505344307cb48d0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.334ex; height:2.176ex;" alt="{\displaystyle 2I}"></li></ul> Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e213bbb69691c65e1391fe16cd79a0029471446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}"> has an <a href="/wiki/Exceptional_object" title="Exceptional object">exceptional</a> irreducible 3-dimensional <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">representation</a> (as the icosahedral rotation group), but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d113353a42f71fc5e7154ddef2257079ab2e25a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{5}}"> does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can also be related to linear groups over the <a href="/wiki/Finite_field" title="Finite field">finite field</a> with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}\cong \operatorname {PSL} (2,5),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>≅</mo> <mi>PSL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}\cong \operatorname {PSL} (2,5),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c393c965fb8eede443a78d31ff58cce76a0bceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.039ex; height:2.843ex;" alt="{\displaystyle A_{5}\cong \operatorname {PSL} (2,5),}"> the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a>, see <a href="/wiki/Projective_linear_group#Action_on_p_points" title="Projective linear group">here</a> for a proof;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{5}\cong \operatorname {PGL} (2,5),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>≅</mo> <mi>PGL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{5}\cong \operatorname {PGL} (2,5),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10a93487ee7f8bfeb874426257bcf1e02bce9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.253ex; height:2.843ex;" alt="{\displaystyle S_{5}\cong \operatorname {PGL} (2,5),}"> the <a href="/wiki/Projective_general_linear_group" class="mw-redirect" title="Projective general linear group">projective general linear group</a>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2I\cong \operatorname {SL} (2,5),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>I</mi> <mo>≅</mo> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2I\cong \operatorname {SL} (2,5),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b7590823236302445cfb8ab3ae5b7a5643b0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.993ex; height:2.843ex;" alt="{\displaystyle 2I\cong \operatorname {SL} (2,5),}"> the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Conjugacy_classes">Conjugacy classes</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=7" title="Edit section: Conjugacy classes">edit</a>]</div> The 120 symmetries fall into 10 conjugacy classes. <table class="wikitable"> <caption><a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a> </caption> <tbody><tr> <th>I </th> <th>additional classes of Ih </th></tr> <tr> <td> <ul><li>identity, order 1</li> <li>12 × rotation by ±72°, order 5, around the 6 axes through the face centers of the dodecahedron</li> <li>12 × rotation by ±144°, order 5, around the 6 axes through the face centers of the dodecahedron</li> <li>20 × rotation by ±120°, order 3, around the 10 axes through vertices of the dodecahedron</li> <li>15 × rotation by 180°, order 2, around the 15 axes through midpoints of edges of the dodecahedron</li></ul> </td> <td> <ul><li>central inversion, order 2</li> <li>12 × rotoreflection by ±36°, order 10, around the 6 axes through the face centers of the dodecahedron</li> <li>12 × rotoreflection by ±108°, order 10, around the 6 axes through the face centers of the dodecahedron</li> <li>20 × rotoreflection by ±60°, order 6, around the 10 axes through the vertices of the dodecahedron</li> <li>15 × reflection, order 2, at 15 planes through edges of the dodecahedron</li></ul> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Subgroups_of_the_full_icosahedral_symmetry_group">Subgroups of the full icosahedral symmetry group</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=8" title="Edit section: Subgroups of the full icosahedral symmetry group">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedral_subgroup_tree.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Icosahedral_subgroup_tree.png/320px-Icosahedral_subgroup_tree.png" decoding="async" width="320" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Icosahedral_subgroup_tree.png/480px-Icosahedral_subgroup_tree.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Icosahedral_subgroup_tree.png/640px-Icosahedral_subgroup_tree.png 2x" data-file-width="840" data-file-height="682" /></a><figcaption>Subgroup relations</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chiral_icosahedral_subgroup_tree.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Chiral_icosahedral_subgroup_tree.png/220px-Chiral_icosahedral_subgroup_tree.png" decoding="async" width="220" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Chiral_icosahedral_subgroup_tree.png/330px-Chiral_icosahedral_subgroup_tree.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/87/Chiral_icosahedral_subgroup_tree.png 2x" data-file-width="431" data-file-height="520" /></a><figcaption>Chiral subgroup relations</figcaption></figure> Each line in the following table represents one class of conjugate (i.e., geometrically equivalent) subgroups. The column "Mult." (multiplicity) gives the number of different subgroups in the conjugacy class. Explanation of colors: green = the groups that are generated by reflections, red = the chiral (orientation-preserving) groups, which contain only rotations. The groups are described geometrically in terms of the dodecahedron. The abbreviation "h.t.s.(edge)" means "halfturn swapping this edge with its opposite edge", and similarly for "face" and "vertex". <table class="wikitable sortable"> <tbody><tr> <th><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schön.</a></th> <th colspan="2"><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter</a></th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a></th> <th><a href="/wiki/Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">H-M</a></th> <th><a href="/wiki/List_of_small_groups" title="List of small groups">Structure</a></th> <th><a href="/wiki/Cycle_diagram" class="mw-redirect" title="Cycle diagram">Cyc.</a></th> <th><a href="/wiki/Symmetry_number" title="Symmetry number">Order</a></th> <th><a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">Index</a></th> <th>Mult.</th> <th>Description </th></tr> <tr align="center" bgcolor="#e0f0f0"> <td>Ih</td> <td>[5,3]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*532</td> <td>532/m</td> <td><a href="/wiki/Alternating_group" title="Alternating group">A5</a>×Z2</td> <td></td> <td>120</td> <td>1</td> <td>1</td> <td>full group </td></tr> <tr align="center" bgcolor="#e0f0f0"> <td>D2h</td> <td>[2,2]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*222</td> <td>mmm</td> <td><a href="/wiki/Dihedral_group" title="Dihedral group">D4</a>×D2=D23</td> <td><a href="/wiki/File:GroupDiagramMiniC2x3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/25px-GroupDiagramMiniC2x3.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/38px-GroupDiagramMiniC2x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/50px-GroupDiagramMiniC2x3.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>8</td> <td>15</td> <td>5</td> <td>fixing two opposite edges, possibly swapping them </td></tr> <tr align="center" bgcolor="#e0f0f0"> <td>C5v</td> <td>[5]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*55</td> <td>5m</td> <td>D10</td> <td><a href="/wiki/File:GroupDiagramMiniD10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/25px-GroupDiagramMiniD10.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/38px-GroupDiagramMiniD10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/50px-GroupDiagramMiniD10.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>10</td> <td>12</td> <td>6</td> <td>fixing a face </td></tr> <tr align="center" bgcolor="#e0f0f0"> <td>C3v</td> <td>[3]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*33</td> <td>3m</td> <td>D6=S3</td> <td><a href="/wiki/File:GroupDiagramMiniD6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/25px-GroupDiagramMiniD6.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/38px-GroupDiagramMiniD6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/50px-GroupDiagramMiniD6.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>6</td> <td>20</td> <td>10</td> <td>fixing a vertex </td></tr> <tr align="center" bgcolor="#e0f0f0"> <td>C2v</td> <td>[2]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*22</td> <td>2mm</td> <td>D4=D22</td> <td><a href="/wiki/File:GroupDiagramMiniD4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/25px-GroupDiagramMiniD4.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/38px-GroupDiagramMiniD4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/50px-GroupDiagramMiniD4.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>4</td> <td>30</td> <td>15</td> <td>fixing an edge </td></tr> <tr align="center" bgcolor="#e0f0f0"> <td>Cs</td> <td>[ ]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>*</td> <td>2 or m</td> <td>D2</td> <td><a href="/wiki/File:GroupDiagramMiniC2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/25px-GroupDiagramMiniC2.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/38px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/50px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>2</td> <td>60</td> <td>15</td> <td>reflection swapping two endpoints of an edge </td></tr> <tr align="center" bgcolor="#f0f0e0"> <td>Th</td> <td>[3+,4]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>3*2</td> <td>m3</td> <td>A4×Z2</td> <td><a href="/wiki/File:GroupDiagramMiniA4xC2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/GroupDiagramMiniA4xC2.png/25px-GroupDiagramMiniA4xC2.png" decoding="async" width="25" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/GroupDiagramMiniA4xC2.png/38px-GroupDiagramMiniA4xC2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/GroupDiagramMiniA4xC2.png/50px-GroupDiagramMiniA4xC2.png 2x" data-file-width="630" data-file-height="497" /></a></td> <td>24</td> <td>5</td> <td>5</td> <td>pyritohedral group </td></tr> <tr align="center" bgcolor="#f0f0e0"> <td>D5d</td> <td>[2+,10]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c6/CDel_10.png" decoding="async" width="10" height="23" class="mw-file-element" data-file-width="10" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>2*5</td> <td>10m2</td> <td>D20=Z2×D10</td> <td><a href="/wiki/File:GroupDiagramMiniD20.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/25px-GroupDiagramMiniD20.png" decoding="async" width="25" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/38px-GroupDiagramMiniD20.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/GroupDiagramMiniD20.png/50px-GroupDiagramMiniD20.png 2x" data-file-width="749" data-file-height="817" /></a></td> <td>20</td> <td>6</td> <td>6</td> <td>fixing two opposite faces, possibly swapping them </td></tr> <tr align="center" bgcolor="#f0f0e0"> <td>D3d</td> <td>[2+,6]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>2*3</td> <td>3m</td> <td>D12=Z2×D6</td> <td><a href="/wiki/File:GroupDiagramMiniD12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/25px-GroupDiagramMiniD12.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/38px-GroupDiagramMiniD12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/50px-GroupDiagramMiniD12.svg.png 2x" data-file-width="64" data-file-height="64" /></a></td> <td>12</td> <td>10</td> <td>10</td> <td>fixing two opposite vertices, possibly swapping them </td></tr> <tr align="center" bgcolor="#f0f0e0"> <td>D1d = C2h</td> <td>[2+,2]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></td> <td>2*</td> <td>2/m</td> <td>D4=<a href="/wiki/Cyclic_group" title="Cyclic group">Z2</a>×D2</td> <td><a href="/wiki/File:GroupDiagramMiniD4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/25px-GroupDiagramMiniD4.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/38px-GroupDiagramMiniD4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/50px-GroupDiagramMiniD4.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>4</td> <td>30</td> <td>15</td> <td>halfturn around edge midpoint, plus central inversion </td></tr> <tr align="center" bgcolor="#e0e0e0"> <td>S10</td> <td>[2+,10+]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/e/e2/CDel_node_h4.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c6/CDel_10.png" decoding="async" width="10" height="23" class="mw-file-element" data-file-width="10" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>5×</td> <td>5</td> <td>Z10=Z2×Z5</td> <td><a href="/wiki/File:GroupDiagramMiniC10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/25px-GroupDiagramMiniC10.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/38px-GroupDiagramMiniC10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/50px-GroupDiagramMiniC10.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>10</td> <td>12</td> <td>6</td> <td>rotations of a face, plus central inversion </td></tr> <tr align="center" bgcolor="#e0e0e0"> <td>S6</td> <td>[2+,6+]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/e/e2/CDel_node_h4.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>3×</td> <td>3</td> <td>Z6=Z2×Z3</td> <td><a href="/wiki/File:GroupDiagramMiniC6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/25px-GroupDiagramMiniC6.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/38px-GroupDiagramMiniC6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/50px-GroupDiagramMiniC6.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>6</td> <td>20</td> <td>10</td> <td>rotations about a vertex, plus central inversion </td></tr> <tr align="center" bgcolor="#e0e0e0"> <td>S2</td> <td>[2+,2+]</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/e/e2/CDel_node_h4.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>×</td> <td>1</td> <td>Z2</td> <td><a href="/wiki/File:GroupDiagramMiniC2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/25px-GroupDiagramMiniC2.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/38px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/50px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>2</td> <td>60</td> <td>1</td> <td>central inversion </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>I</td> <td>[5,3]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>532</td> <td>532</td> <td>A5</td> <td></td> <td>60</td> <td>2</td> <td>1</td> <td>all rotations </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>T</td> <td>[3,3]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>332</td> <td>332</td> <td>A4</td> <td><a href="/wiki/File:GroupDiagramMiniA4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/25px-GroupDiagramMiniA4.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/38px-GroupDiagramMiniA4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/50px-GroupDiagramMiniA4.svg.png 2x" data-file-width="64" data-file-height="64" /></a></td> <td>12</td> <td>10</td> <td>5</td> <td>rotations of a contained tetrahedron </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>D5</td> <td>[2,5]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>522</td> <td>522</td> <td>D10</td> <td><a href="/wiki/File:GroupDiagramMiniD10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/25px-GroupDiagramMiniD10.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/38px-GroupDiagramMiniD10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/50px-GroupDiagramMiniD10.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>10</td> <td>12</td> <td>6</td> <td>rotations around the center of a face, and h.t.s.(face) </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>D3</td> <td>[2,3]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>322</td> <td>322</td> <td>D6=S3</td> <td><a href="/wiki/File:GroupDiagramMiniD6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/25px-GroupDiagramMiniD6.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/38px-GroupDiagramMiniD6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/50px-GroupDiagramMiniD6.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>6</td> <td>20</td> <td>10</td> <td>rotations around a vertex, and h.t.s.(vertex) </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>D2</td> <td>[2,2]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>222</td> <td>222</td> <td>D4=Z22</td> <td><a href="/wiki/File:GroupDiagramMiniD4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/25px-GroupDiagramMiniD4.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/38px-GroupDiagramMiniD4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/50px-GroupDiagramMiniD4.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>4</td> <td>30</td> <td>5</td> <td>halfturn around edge midpoint, and h.t.s.(edge) </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>C5</td> <td>[5]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>55</td> <td>5</td> <td>Z5</td> <td><a href="/wiki/File:GroupDiagramMiniC5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/25px-GroupDiagramMiniC5.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/38px-GroupDiagramMiniC5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/50px-GroupDiagramMiniC5.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>5</td> <td>24</td> <td>6</td> <td>rotations around a face center </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>C3</td> <td>[3]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>33</td> <td>3</td> <td>Z3=A3</td> <td><a href="/wiki/File:GroupDiagramMiniC3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/25px-GroupDiagramMiniC3.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/38px-GroupDiagramMiniC3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/50px-GroupDiagramMiniC3.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>3</td> <td>40</td> <td>10</td> <td>rotations around a vertex </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>C2</td> <td>[2]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>22</td> <td>2</td> <td>Z2</td> <td><a href="/wiki/File:GroupDiagramMiniC2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/25px-GroupDiagramMiniC2.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/38px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/50px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>2</td> <td>60</td> <td>15</td> <td>half-turn around edge midpoint </td></tr> <tr align="center" bgcolor="#f0e0f0"> <td>C1</td> <td>[ ]+</td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></td> <td>11</td> <td>1</td> <td>Z1</td> <td><a href="/wiki/File:GroupDiagramMiniC1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/25px-GroupDiagramMiniC1.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/38px-GroupDiagramMiniC1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/50px-GroupDiagramMiniC1.svg.png 2x" data-file-width="32" data-file-height="32" /></a></td> <td>1</td> <td>120</td> <td>1</td> <td>trivial group </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Vertex_stabilizers">Vertex stabilizers</h4>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=9" title="Edit section: Vertex stabilizers">edit</a>]</div> Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. <ul><li>vertex stabilizers in I give <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> C3</li> <li>vertex stabilizers in Ih give <a href="/wiki/Dihedral_symmetry_in_three_dimensions" title="Dihedral symmetry in three dimensions">dihedral groups</a> D3</li> <li>stabilizers of an opposite pair of vertices in I give dihedral groups D3</li> <li>stabilizers of an opposite pair of vertices in Ih give <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3}\times \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>×</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{3}\times \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c633e910868413a255c185be4e799d2c218ab56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.789ex; height:2.509ex;" alt="{\displaystyle D_{3}\times \pm 1}"></li></ul> <div class="mw-heading mw-heading4"><h4 id="Edge_stabilizers">Edge stabilizers</h4>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=10" title="Edit section: Edge stabilizers">edit</a>]</div> Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate. <ul><li>edges stabilizers in I give cyclic groups Z2</li> <li>edges stabilizers in Ih give <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{2}\times Z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{2}\times Z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e55f4fa0a6e390a7edf3a1e8d141034f2fcb2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.124ex; height:2.509ex;" alt="{\displaystyle Z_{2}\times Z_{2}}"></li> <li>stabilizers of a pair of edges in I give <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{2}\times Z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{2}\times Z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e55f4fa0a6e390a7edf3a1e8d141034f2fcb2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.124ex; height:2.509ex;" alt="{\displaystyle Z_{2}\times Z_{2}}">; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.</li> <li>stabilizers of a pair of edges in Ih give <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{2}\times Z_{2}\times Z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{2}\times Z_{2}\times Z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d953a906608cfbdef7a2b36602498d4cd89df340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.606ex; height:2.509ex;" alt="{\displaystyle Z_{2}\times Z_{2}\times Z_{2}}">; there are 5 of these, given by reflections in 3 perpendicular axes.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Face_stabilizers">Face stabilizers</h4>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=11" title="Edit section: Face stabilizers">edit</a>]</div> Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the <a href="/wiki/Antiprism" title="Antiprism">antiprism</a> they generate. <ul><li>face stabilizers in I give cyclic groups C5</li> <li>face stabilizers in Ih give dihedral groups D5</li> <li>stabilizers of an opposite pair of faces in I give dihedral groups D5</li> <li>stabilizers of an opposite pair of faces in Ih give <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{5}\times \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>×</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{5}\times \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/435f95647411264d39c51d9a5c1c1f61e8dad3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.789ex; height:2.509ex;" alt="{\displaystyle D_{5}\times \pm 1}"></li></ul> <div class="mw-heading mw-heading4"><h4 id="Polyhedron_stabilizers">Polyhedron stabilizers</h4>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=12" title="Edit section: Polyhedron stabilizers">edit</a>]</div> For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I{\stackrel {\sim }{\to }}A_{5}<S_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mover> </mrow> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo><</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I{\stackrel {\sim }{\to }}A_{5}<S_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09e712547d5689ea9234569d52c18c556d1dd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.87ex; height:3.176ex;" alt="{\displaystyle I{\stackrel {\sim }{\to }}A_{5}<S_{5}}">. <ul><li>stabilizers of the inscribed tetrahedra in I are a copy of T</li> <li>stabilizers of the inscribed tetrahedra in Ih are a copy of T</li> <li>stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in I are a copy of T</li> <li>stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in Ih are a copy of Th</li></ul> <div class="mw-heading mw-heading4"><h4 id="Coxeter_group_generators">Coxeter group generators</h4>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=13" title="Edit section: Coxeter group generators">edit</a>]</div> The full icosahedral symmetry group [5,3] (<img src="//upload.wikimedia.org/wikipedia/commons/a/a8/CDel_node_n0.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/CDel_node_n1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/2/20/CDel_node_n2.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" />) of order 120 has generators represented by the reflection matrices R0, R1, R2 below, with relations R02 = R12 = R22 = (R0×R1)5 = (R1×R2)3 = (R0×R2)2 = Identity. The group [5,3]+ (<img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" />) of order 60 is generated by any two of the rotations S0,1, S1,2, S0,2. A <a href="/wiki/Rotoreflection" class="mw-redirect" title="Rotoreflection">rotoreflection</a> of order 10 is generated by V0,1,2, the product of all 3 reflections. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ={\tfrac {{\sqrt {5}}+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ={\tfrac {{\sqrt {5}}+1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422d2d2bb8b6aaa7577a3f39c8fd7e00cf6c488f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.611ex; height:4.176ex;" alt="{\displaystyle \phi ={\tfrac {{\sqrt {5}}+1}{2}}}"> denotes the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>. <table class="wikitable"> <caption>[5,3], <img src="//upload.wikimedia.org/wikipedia/commons/a/a8/CDel_node_n0.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/CDel_node_n1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/2/20/CDel_node_n2.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </caption> <tbody><tr> <th> </th> <th colspan="3">Reflections </th> <th colspan="3">Rotations </th> <th>Rotoreflection </th></tr> <tr> <th>Name </th> <th>R0 </th> <th>R1 </th> <th>R2 </th> <th>S0,1 </th> <th>S1,2 </th> <th>S0,2 </th> <th>V0,1,2 </th></tr> <tr align="center"> <th>Group </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/a/a8/CDel_node_n0.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/CDel_node_n1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/2/20/CDel_node_n2.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c6/CDel_10.png" decoding="async" width="10" height="23" class="mw-file-element" data-file-width="10" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/e/e2/CDel_node_h4.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/70/CDel_node_h2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /> </td></tr> <tr align="center"> <th>Order </th> <td>2</td> <td>2</td> <td>2</td> <td>5</td> <td>3</td> <td>2</td> <td>10 </td></tr> <tr align="center"> <th>Matrix </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81bf2648cef2b0f3cb2f82f0f0bcba89c595af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.498ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" 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</mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a2f124af793abcdec2a757fa54eacd8e12183f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.548ex; margin-bottom: -0.29ex; width:15.409ex; height:10.843ex;" alt="{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi 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src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d98311a57b2d6575867fb3100199be36b656299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.498ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef78ac2c56b1aeecf4b2ee91fa450dd1b0bbc29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.548ex; margin-bottom: -0.29ex; width:15.409ex; height:10.843ex;" alt="{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c62a981424dfb7bce1a413d58b0ee62a9322924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.548ex; margin-bottom: -0.29ex; width:15.409ex; height:10.843ex;" alt="{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ecfba65b611468342e533ba54ae7b9f240f3bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.776ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f358516e6f6d9c29e3ca1c9f6bac53e18019636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.548ex; margin-bottom: -0.29ex; width:15.409ex; height:10.843ex;" alt="{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}"> </td></tr> <tr align="center"> <th> </th> <td>(1,0,0)n </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5badffa1c426d6b1384ccf2a5734034503b7aff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.947ex; height:3.509ex;" alt="{\displaystyle ({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})}">n </td> <td>(0,1,0)n </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,-1,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,-1,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7c920f35e51d8d621783f0c070e9f375c5e07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.843ex;" alt="{\displaystyle (0,-1,\phi )}">axis </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\phi ,0,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ϕ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\phi ,0,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db7f81f7b2df759441bdcaffcc9e37d7b2e96c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.813ex; height:2.843ex;" alt="{\displaystyle (1-\phi ,0,\phi )}">axis </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b35543ef5f96df2e9aea6cced9bdd57dab7e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,1)}">axis </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Fundamental_domain">Fundamental domain</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=14" title="Edit section: Fundamental domain">edit</a>]</div> <a href="/wiki/Fundamental_domain" title="Fundamental domain">Fundamental domains</a> for the icosahedral rotation group and the full icosahedral group are given by: <table class="wikitable" width="580"> <tbody><tr align="center" valign="top"> <td><a href="/wiki/File:Sphere_symmetry_group_i.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/200px-Sphere_symmetry_group_i.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/300px-Sphere_symmetry_group_i.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Sphere_symmetry_group_i.png/400px-Sphere_symmetry_group_i.png 2x" data-file-width="985" data-file-height="987" /></a> Icosahedral rotation group I </td> <td><a href="/wiki/File:Sphere_symmetry_group_ih.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/200px-Sphere_symmetry_group_ih.png" decoding="async" width="200" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/300px-Sphere_symmetry_group_ih.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/400px-Sphere_symmetry_group_ih.png 2x" data-file-width="671" data-file-height="617" /></a> Full icosahedral group Ih </td> <td><a href="/wiki/File:Disdyakistriacontahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/180px-Disdyakistriacontahedron.jpg" decoding="async" width="180" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/270px-Disdyakistriacontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/360px-Disdyakistriacontahedron.jpg 2x" data-file-width="812" data-file-height="847" /></a> Faces of <a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a> are the fundamental domain </td></tr></tbody></table> In the <a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a> one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface. <div class="mw-heading mw-heading2"><h2 id="Polyhedra_with_icosahedral_symmetry">Polyhedra with icosahedral symmetry</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=15" title="Edit section: Polyhedra with icosahedral symmetry">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Solids_with_icosahedral_symmetry" title="Solids with icosahedral symmetry">Solids with icosahedral symmetry</a></div>Examples of other <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> with icosahedral symmetry include the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> (the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> of the icosahedron) and the <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a>. <div class="mw-heading mw-heading3"><h3 id="Chiral_polyhedra">Chiral polyhedra</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=16" title="Edit section: Chiral polyhedra">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th>Class </th> <th>Symbols </th> <th>Picture </th></tr> <tr align="center"> <th><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean</a> </th> <th><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">sr{5,3}</a> <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </th> <td><a href="/wiki/File:Snubdodecahedronccw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Snubdodecahedronccw.jpg/50px-Snubdodecahedronccw.jpg" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Snubdodecahedronccw.jpg/75px-Snubdodecahedronccw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Snubdodecahedronccw.jpg/100px-Snubdodecahedronccw.jpg 2x" data-file-width="853" data-file-height="866" /></a> </td></tr> <tr align="center"> <th><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan</a> </th> <th><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">V3.3.3.3.5</a> <img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </th> <td><a href="/wiki/File:Pentagonalhexecontahedronccw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Pentagonalhexecontahedronccw.jpg/50px-Pentagonalhexecontahedronccw.jpg" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Pentagonalhexecontahedronccw.jpg/75px-Pentagonalhexecontahedronccw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Pentagonalhexecontahedronccw.jpg/100px-Pentagonalhexecontahedronccw.jpg 2x" data-file-width="834" data-file-height="851" /></a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Full_icosahedral_symmetry">Full icosahedral symmetry</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=17" title="Edit section: Full icosahedral symmetry">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th colspan="1"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a></th> <th colspan="2"><a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedra</a> </th> <th colspan="5"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a> </th></tr> <tr align="center"> <td><a href="/wiki/File:Dodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/50px-Dodecahedron.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/75px-Dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/100px-Dodecahedron.svg.png 2x" data-file-width="300" data-file-height="300" /></a> <a href="/wiki/Dodecahedron" title="Dodecahedron">{5,3}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:SmallStellatedDodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SmallStellatedDodecahedron.jpg/50px-SmallStellatedDodecahedron.jpg" decoding="async" width="50" height="53" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SmallStellatedDodecahedron.jpg/75px-SmallStellatedDodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SmallStellatedDodecahedron.jpg/100px-SmallStellatedDodecahedron.jpg 2x" data-file-width="770" data-file-height="822" /></a> <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">{5/2,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:GreatStellatedDodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/GreatStellatedDodecahedron.jpg/50px-GreatStellatedDodecahedron.jpg" decoding="async" width="50" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/GreatStellatedDodecahedron.jpg/75px-GreatStellatedDodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/GreatStellatedDodecahedron.jpg/100px-GreatStellatedDodecahedron.jpg 2x" data-file-width="853" data-file-height="794" /></a> <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">{5/2,3}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Truncateddodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Truncateddodecahedron.jpg/50px-Truncateddodecahedron.jpg" decoding="async" width="50" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Truncateddodecahedron.jpg/75px-Truncateddodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Truncateddodecahedron.jpg/100px-Truncateddodecahedron.jpg 2x" data-file-width="876" data-file-height="802" /></a> <a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">t{5,3}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Truncatedicosahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Truncatedicosahedron.jpg/50px-Truncatedicosahedron.jpg" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Truncatedicosahedron.jpg/75px-Truncatedicosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Truncatedicosahedron.jpg/100px-Truncatedicosahedron.jpg 2x" data-file-width="878" data-file-height="874" /></a> <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">t{3,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td><a href="/wiki/File:Icosidodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Icosidodecahedron.svg/50px-Icosidodecahedron.svg.png" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Icosidodecahedron.svg/75px-Icosidodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Icosidodecahedron.svg/100px-Icosidodecahedron.svg.png 2x" data-file-width="841" data-file-height="861" /></a> <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">r{3,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Rhombicosidodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Rhombicosidodecahedron.jpg/50px-Rhombicosidodecahedron.jpg" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Rhombicosidodecahedron.jpg/75px-Rhombicosidodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Rhombicosidodecahedron.jpg/100px-Rhombicosidodecahedron.jpg 2x" data-file-width="858" data-file-height="871" /></a> <a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rr{3,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td><a href="/wiki/File:Truncatedicosidodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Truncatedicosidodecahedron.jpg/50px-Truncatedicosidodecahedron.jpg" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Truncatedicosidodecahedron.jpg/75px-Truncatedicosidodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Truncatedicosidodecahedron.jpg/100px-Truncatedicosidodecahedron.jpg 2x" data-file-width="868" data-file-height="869" /></a> <a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">tr{3,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td></tr> <tr align="center"> <th>Platonic solid</th> <th colspan="2">Kepler–Poinsot polyhedra </th> <th colspan="5"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a> </th></tr> <tr align="center"> <td><a href="/wiki/File:Icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/50px-Icosahedron.svg.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/75px-Icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/100px-Icosahedron.svg.png 2x" data-file-width="512" data-file-height="492" /></a> <a href="/wiki/Icosahedron" title="Icosahedron">{3,5}</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> = <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td><a href="/wiki/File:GreatDodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/GreatDodecahedron.jpg/50px-GreatDodecahedron.jpg" decoding="async" width="50" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/GreatDodecahedron.jpg/75px-GreatDodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/GreatDodecahedron.jpg/100px-GreatDodecahedron.jpg 2x" data-file-width="752" data-file-height="814" /></a> <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">{5,5/2}</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> = <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td><a href="/wiki/File:GreatIcosahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/GreatIcosahedron.jpg/50px-GreatIcosahedron.jpg" decoding="async" width="50" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/GreatIcosahedron.jpg/75px-GreatIcosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/GreatIcosahedron.jpg/100px-GreatIcosahedron.jpg 2x" data-file-width="831" data-file-height="848" /></a> <a href="/wiki/Great_icosahedron" title="Great icosahedron">{3,5/2}</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> = <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/67/CDel_5-2.png" decoding="async" width="14" height="23" class="mw-file-element" data-file-width="14" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td><a href="/wiki/File:Triakisicosahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/50px-Triakisicosahedron.jpg" decoding="async" width="50" height="52" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/75px-Triakisicosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/100px-Triakisicosahedron.jpg 2x" data-file-width="819" data-file-height="849" /></a> <a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">V3.10.10</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Pentakisdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/50px-Pentakisdodecahedron.jpg" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/75px-Pentakisdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/100px-Pentakisdodecahedron.jpg 2x" data-file-width="844" data-file-height="843" /></a> <a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">V5.6.6</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Rhombictriacontahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/50px-Rhombictriacontahedron.svg.png" decoding="async" width="50" height="55" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/75px-Rhombictriacontahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/100px-Rhombictriacontahedron.svg.png 2x" data-file-width="560" data-file-height="620" /></a> <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">V3.5.3.5</a> <img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Deltoidalhexecontahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/50px-Deltoidalhexecontahedron.jpg" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/75px-Deltoidalhexecontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/100px-Deltoidalhexecontahedron.jpg 2x" data-file-width="854" data-file-height="843" /></a> <a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">V3.4.5.4</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td><a href="/wiki/File:Disdyakistriacontahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/50px-Disdyakistriacontahedron.jpg" decoding="async" width="50" height="52" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/75px-Disdyakistriacontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/100px-Disdyakistriacontahedron.jpg 2x" data-file-width="812" data-file-height="847" /></a> <a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">V4.6.10</a> <img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Other_objects_with_icosahedral_symmetry">Other objects with icosahedral symmetry</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=18" title="Edit section: Other objects with icosahedral symmetry">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader">Examples of icosahedral symmetry</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Circogoniaicosahedra_ekw.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/0/02/Circogoniaicosahedra_ekw.jpg" decoding="async" width="200" height="223" class="mw-file-element" data-file-width="157" data-file-height="175" /></a></div><div class="thumbcaption"><a href="/w/index.php?title=Circogonia_icosahedra&action=edit&redlink=1" class="new" title="Circogonia icosahedra (page does not exist)">Circogonia icosahedra</a>, a <a href="/wiki/Radiolarian" class="mw-redirect" title="Radiolarian">Radiolarian</a></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Adenovirus_3D_schematic.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Adenovirus_3D_schematic.png/200px-Adenovirus_3D_schematic.png" decoding="async" width="200" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Adenovirus_3D_schematic.png/300px-Adenovirus_3D_schematic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Adenovirus_3D_schematic.png/400px-Adenovirus_3D_schematic.png 2x" data-file-width="1512" data-file-height="1726" /></a></div><div class="thumbcaption">Capsid of an <a href="/wiki/Adenoviridae" title="Adenoviridae">Adenovirus</a></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Dodecaborate(12)-dianion-from-xtal-3D-bs-17.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png/200px-Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png" decoding="async" width="200" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png/300px-Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png/400px-Dodecaborate%2812%29-dianion-from-xtal-3D-bs-17.png 2x" data-file-width="2122" data-file-height="2380" /></a></div><div class="thumbcaption">The <a href="/wiki/Dodecaborate" title="Dodecaborate">dodecaborate</a> ion [B12H12]2−</div></div></div></div></div> <ul><li><a href="/wiki/Barth_surface" title="Barth surface">Barth surfaces</a></li> <li><a href="/wiki/Virus_structure" class="mw-redirect" title="Virus structure">Virus structure</a>, and <a href="/wiki/Capsid" title="Capsid">Capsid</a></li> <li>In chemistry, the <a href="/wiki/Dodecaborate" title="Dodecaborate">dodecaborate</a> ion ([B12H12]2−) and the <a href="/wiki/Dodecahedrane" title="Dodecahedrane">dodecahedrane</a> molecule (C20H20)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Liquid_crystals_with_icosahedral_symmetry">Liquid crystals with icosahedral symmetry</h3>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=19" title="Edit section: Liquid crystals with icosahedral symmetry">edit</a>]</div> For the intermediate material phase called <a href="/wiki/Liquid_crystals" class="mw-redirect" title="Liquid crystals">liquid crystals</a> the existence of icosahedral symmetry was proposed by <a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">H. Kleinert</a> and K. Maki<a href="#cite_note-2">[2]</a> and its structure was first analyzed in detail in that paper. See the review article <a rel="nofollow" class="external text" href="http://chemgroups.northwestern.edu/seideman/Publications/The%20liquid-crystalline%20blue%20phases.pdf">here</a>. In aluminum, the icosahedral structure was discovered experimentally three years after this by <a href="/wiki/Dan_Shechtman" title="Dan Shechtman">Dan Shechtman</a>, which earned him the Nobel Prize in 2011. <div class="mw-heading mw-heading2"><h2 id="Related_geometries">Related geometries</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=20" title="Edit section: Related geometries">edit</a>]</div> Icosahedral symmetry is equivalently the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a> PSL(2,5), and is the symmetry group of the <a href="/wiki/Modular_curve" title="Modular curve">modular curve</a> X(5), and more generally PSL(2,p) is the symmetry group of the modular curve X(p). The modular curve X(5) is geometrically a dodecahedron with a cusp at the center of each polygonal face, which demonstrates the symmetry group. This geometry, and associated symmetry group, was studied by <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> as the <a href="/wiki/Monodromy_group" class="mw-redirect" title="Monodromy group">monodromy groups</a> of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a <a href="/wiki/Belyi_function" class="mw-redirect" title="Belyi function">Belyi function</a>) – the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over 0 and 1; the degree of the covering (number of sheets) equals 5. This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of the <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a>, with the theory given in the famous (<a href="#CITEREFKlein1888">Klein 1888</a>); a modern exposition is given in (<a href="#CITEREFTóth2002">Tóth 2002</a>, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i76mmyvDHYUC&pg=PA66">p. 66</a>). Klein's investigations continued with his discovery of order 7 and order 11 symmetries in (<a href="#CITEREFKlein1878">Klein 1878</a>) and (<a href="#CITEREFKlein1879">Klein 1879</a>) (and associated coverings of degree 7 and 11) and <a href="/wiki/Dessins_d%27enfants" class="mw-redirect" title="Dessins d'enfants">dessins d'enfants</a>, the first yielding the <a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a>, whose associated geometry has a tiling by 24 heptagons (with a cusp at the center of each). Similar geometries occur for PSL(2,n) and more general groups for other modular curves. More exotically, there are special connections between the groups PSL(2,5) (order 60), <a href="/wiki/PSL(2,7)" title="PSL(2,7)">PSL(2,7)</a> (order 168) and PSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron (genus 0), PSL(2,7) of the <a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a> (genus 3), and PSL(2,11) the <a href="/wiki/Buckyball_surface" class="mw-redirect" title="Buckyball surface">buckyball surface</a> (genus 70). These groups form a "<a href="/wiki/ADE_classification#Trinities" title="ADE classification">trinity</a>" in the sense of <a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Vladimir Arnold</a>, which gives a framework for the various relationships; see <a href="/wiki/ADE_classification#Trinities" title="ADE classification">trinities</a> for details. There is a close relationship to other <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=21" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">Tetrahedral symmetry</a></li> <li><a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">Octahedral symmetry</a></li> <li><a href="/wiki/Binary_icosahedral_group" title="Binary icosahedral group">Binary icosahedral group</a></li> <li><a href="/wiki/Icosian_calculus" title="Icosian calculus">Icosian calculus</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=22" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSir_William_Rowan_Hamilton1856" class="citation cs2"><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Sir William Rowan Hamilton</a> (1856), <a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf">"Memorandum respecting a new System of Roots of Unity"</a> (PDF), <a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a>, 12: 446</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=Memorandum+respecting+a+new+System+of+Roots+of+Unity&rft.volume=12&rft.pages=446&rft.date=1856&rft.au=Sir+William+Rowan+Hamilton&rft_id=http%3A%2F%2Fwww.maths.tcd.ie%2Fpub%2FHistMath%2FPeople%2FHamilton%2FIcosian%2FNewSys.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinert,_H.Maki,_K.1981" class="citation journal cs1"><a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Kleinert, H.</a> & Maki, K. (1981). <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/75/75.pdf">"Lattice Textures in Cholesteric Liquid Crystals"</a> (PDF). Fortschritte der Physik. 29 (5): 219–259. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fprop.19810290503">10.1002/prop.19810290503</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fortschritte+der+Physik&rft.atitle=Lattice+Textures+in+Cholesteric+Liquid+Crystals&rft.volume=29&rft.issue=5&rft.pages=219-259&rft.date=1981&rft_id=info%3Adoi%2F10.1002%2Fprop.19810290503&rft.au=Kleinert%2C+H.&rft.au=Maki%2C+K.&rft_id=http%3A%2F%2Fwww.physik.fu-berlin.de%2F~kleinert%2F75%2F75.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1878" class="citation journal cs1"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, F.</a> (1878). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1635511">"Ueber die Transformation siebenter Ordnung der elliptischen Functionen"</a> [On the order-seven transformation of elliptic functions]. Mathematische Annalen. 14 (3): 428–471. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01677143">10.1007/BF01677143</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121407539">121407539</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Ueber+die+Transformation+siebenter+Ordnung+der+elliptischen+Functionen&rft.volume=14&rft.issue=3&rft.pages=428-471&rft.date=1878&rft_id=info%3Adoi%2F10.1007%2FBF01677143&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121407539%23id-name%3DS2CID&rft.aulast=Klein&rft.aufirst=F.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1635511&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"> Translated in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevy1999" class="citation book cs1">Levy, Silvio, ed. (1999). <a rel="nofollow" class="external text" href="http://www.msri.org/communications/books/Book35/index.html">The Eightfold Way</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-66066-2" title="Special:BookSources/978-0-521-66066-2"><bdi>978-0-521-66066-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1722410">1722410</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Eightfold+Way&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-66066-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1722410%23id-name%3DMR&rft_id=http%3A%2F%2Fwww.msri.org%2Fcommunications%2Fbooks%2FBook35%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1879" class="citation cs2"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, F.</a> (1879), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1642598">"Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh order transformation of elliptic functions)"</a>, Mathematische Annalen, 15 (3–4): 533–555, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02086276">10.1007/BF02086276</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120316938">120316938</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Ueber+die+Transformation+elfter+Ordnung+der+elliptischen+Functionen+%28On+the+eleventh+order+transformation+of+elliptic+functions%29&rft.volume=15&rft.issue=3%E2%80%934&rft.pages=533-555&rft.date=1879&rft_id=info%3Adoi%2F10.1007%2FBF02086276&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120316938%23id-name%3DS2CID&rft.aulast=Klein&rft.aufirst=F.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1642598&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988">, collected as pp. 140–165 in <a rel="nofollow" class="external text" href="http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_KLEIN__3">Oeuvres, Tome 3</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1888" class="citation cs2">Klein, Felix (1888), Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Trübner & Co., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-49528-0" title="Special:BookSources/0-486-49528-0"><bdi>0-486-49528-0</bdi>trans</a>. George Gavin Morrice</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+Icosahedron+and+the+Solution+of+Equations+of+the+Fifth+Degree&rft.pub=Tr%C3%BCbner+%26+Co.&rft.date=1888&rft.isbn=0-486-49528-0&rft.aulast=Klein&rft.aufirst=Felix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: postscript (<a href="/wiki/Category:CS1_maint:_postscript" title="Category:CS1 maint: postscript">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTóth2002" class="citation cs2">Tóth, Gábor (2002), Finite Möbius groups, minimal immersions of spheres, and moduli</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+M%C3%B6bius+groups%2C+minimal+immersions+of+spheres%2C+and+moduli&rft.date=2002&rft.aulast=T%C3%B3th&rft.aufirst=G%C3%A1bor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"></li> <li>Peter R. Cromwell, Polyhedra (1997), p. 296</li> <li>The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-220-5" title="Special:BookSources/978-1-56881-220-5">978-1-56881-220-5</a></li> <li>Kaleidoscopes: Selected Writings of <a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">H.S.M. Coxeter</a>, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-01003-6" title="Special:BookSources/978-0-471-01003-6">978-0-471-01003-6</a> <a rel="nofollow" class="external autonumber" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a></li> <li><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">N.W. Johnson</a>: Geometries and Transformations, (2018) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-10340-5" title="Special:BookSources/978-1-107-10340-5">978-1-107-10340-5</a> Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Icosahedral_symmetry&action=edit&section=23" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/IcosahedralGroup.html">"Icosahedral group"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Icosahedral+group&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FIcosahedralGroup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedral+symmetry" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://schmidt.nuigalway.ie/subgroups/h3.pdf">THE SUBGROUPS OF W(H3)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210830211728/http://schmidt.nuigalway.ie/subgroups/h3.pdf">Archived</a> 2021-08-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (<a rel="nofollow" class="external text" href="http://schmidt.nuigalway.ie/subgroups/cox.html">Subgroups of other Coxeter groups</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200802022826/http://schmidt.nuigalway.ie/subgroups/cox.html">Archived</a> 2020-08-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>) Gotz Pfeiffer</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Icosahedral_symmetry&oldid=1245738611">https://en.wikipedia.org/w/index.php?title=Icosahedral_symmetry&oldid=1245738611</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Finite_groups" title="Category:Finite groups">Finite groups</a></li><li><a href="/wiki/Category:Rotational_symmetry" title="Category:Rotational symmetry">Rotational symmetry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_May_2020" title="Category:Articles lacking in-text citations from May 2020">Articles lacking in-text citations from May 2020</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:CS1_maint:_postscript" title="Category:CS1 maint: postscript">CS1 maint: postscript</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 September 2024, at 19:51 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Icosahedral symmetry - Wikipedia