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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Classification system for crystals</div> <p>In <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>, a <b>crystallographic point group</b> is a <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">three dimensional point group</a> whose symmetry operations are compatible with a three dimensional crystallographic <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a>. According to the <a href="/wiki/Crystallographic_restriction_theorem" title="Crystallographic restriction theorem">crystallographic restriction</a> it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are one-and-the-same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by <a href="/wiki/Johann_F._C._Hessel" title="Johann F. C. Hessel">Johann Friedrich Christian Hessel</a> from a consideration of observed crystal forms. </p><p>In the classification of crystals, to each <a href="/wiki/Space_group" title="Space group">space group</a> is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations. That is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the <b>(geometric) crystal class</b> of the crystal. </p><p>The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including <a href="/wiki/Crystal_optics" title="Crystal optics">optical properties</a> such as <a href="/wiki/Birefringence" title="Birefringence">birefringency</a>, or electro-optical features such as the <a href="/wiki/Pockels_effect" title="Pockels effect">Pockels effect</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=1" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, <a href="/wiki/Mineralogist" class="mw-redirect" title="Mineralogist">mineralogists</a>, and <a href="/wiki/Physicists" class="mw-redirect" title="Physicists">physicists</a>. </p><p>For the correspondence of the two systems below, see <b><a href="/wiki/Crystal_system" title="Crystal system">crystal system</a></b>. </p> <div class="mw-heading mw-heading3"><h3 id="Schoenflies_notation">Schoenflies notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=2" title="Edit section: Schoenflies notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoenflies notation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">Point groups in three dimensions</a></div> <p>In <a href="/wiki/Arthur_Moritz_Schoenflies" title="Arthur Moritz Schoenflies">Schoenflies</a> notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: </p> <ul><li><i>C<sub>n</sub></i> (for <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a>) indicates that the group has an <i>n</i>-fold rotation axis. <i>C<sub>nh</sub></i> is <i>C<sub>n</sub></i> with the addition of a mirror (reflection) plane perpendicular to the <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a>. <i>C<sub>nv</sub></i> is <i>C<sub>n</sub></i> with the addition of n mirror planes parallel to the axis of rotation.</li> <li><i>S<sub>2n</sub></i> (for <i>Spiegel</i>, German for <a href="/wiki/Mirror" title="Mirror">mirror</a>) denotes a group with only a <i>2n</i>-fold <a href="/wiki/Improper_rotation" title="Improper rotation">rotation-reflection axis</a>.</li> <li><i>D<sub>n</sub></i> (for <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a>, or two-sided) indicates that the group has an <i>n</i>-fold rotation axis plus <i>n</i> twofold axes perpendicular to that axis. <i>D<sub>nh</sub></i> has, in addition, a mirror plane perpendicular to the <i>n</i>-fold axis. <i>D<sub>nd</sub></i> has, in addition to the elements of <i>D<sub>n</sub></i>, mirror planes parallel to the <i>n</i>-fold axis.</li> <li>The letter <i>T</i> (for <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>) indicates that the group has the symmetry of a tetrahedron. <i>T<sub>d</sub></i> includes <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotation</a> operations, <i>T</i> excludes improper rotation operations, and <i>T<sub>h</sub></i> is <i>T</i> with the addition of an inversion.</li> <li>The letter <i>O</i> (for <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>) indicates that the group has the symmetry of an octahedron, with (<i>O<sub>h</sub></i>) or without (<i>O</i>) improper operations (those that change handedness).</li></ul> <p>Due to the <a href="/wiki/Crystallographic_restriction_theorem" title="Crystallographic restriction theorem">crystallographic restriction theorem</a>, <i>n</i> = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space. </p> <table class="wikitable"> <tbody><tr> <th>n </th> <th>1 </th> <th>2 </th> <th>3 </th> <th>4 </th> <th>6 </th></tr> <tr> <td><i>C<sub>n</sub></i> </td> <td><i>C<sub>1</sub></i> </td> <td><i>C<sub>2</sub></i> </td> <td><i>C<sub>3</sub></i> </td> <td><i>C<sub>4</sub></i> </td> <td><i>C<sub>6</sub></i> </td></tr> <tr> <td><i>C<sub>nv</sub></i> </td> <td><i>C<sub>1v</sub></i>=<i>C<sub>1h</sub></i> </td> <td><i>C<sub>2v</sub></i> </td> <td><i>C<sub>3v</sub></i> </td> <td><i>C<sub>4v</sub></i> </td> <td><i>C<sub>6v</sub></i> </td></tr> <tr> <td><i>C<sub>nh</sub></i> </td> <td><i>C<sub>1h</sub></i> </td> <td><i>C<sub>2h</sub></i> </td> <td><i>C<sub>3h</sub></i> </td> <td><i>C<sub>4h</sub></i> </td> <td><i>C<sub>6h</sub></i> </td></tr> <tr> <td><i>D<sub>n</sub></i> </td> <td><i>D<sub>1</sub></i>=<i>C<sub>2</sub></i> </td> <td><i>D<sub>2</sub></i> </td> <td><i>D<sub>3</sub></i> </td> <td><i>D<sub>4</sub></i> </td> <td><i>D<sub>6</sub></i> </td></tr> <tr> <td><i>D<sub>nh</sub></i> </td> <td><i>D<sub>1h</sub></i>=<i>C<sub>2v</sub></i> </td> <td><i>D<sub>2h</sub></i> </td> <td><i>D<sub>3h</sub></i> </td> <td><i>D<sub>4h</sub></i> </td> <td><i>D<sub>6h</sub></i> </td></tr> <tr> <td><i>D<sub>nd</sub></i> </td> <td><i>D<sub>1d</sub></i>=<i>C<sub>2h</sub></i> </td> <td><i>D<sub>2d</sub></i> </td> <td><i>D<sub>3d</sub></i> </td> <td style="background:silver"><i>D<sub>4d</sub></i> </td> <td style="background:silver"><i>D<sub>6d</sub></i> </td></tr> <tr> <td><i>S<sub>2n</sub></i> </td> <td><i>S<sub>2</sub></i> </td> <td><i>S<sub>4</sub></i> </td> <td><i>S<sub>6</sub></i> </td> <td style="background:silver"><i>S<sub>8</sub></i> </td> <td style="background:silver"><i>S<sub>12</sub></i> </td></tr></tbody></table> <p><i>D<sub>4d</sub></i> and <i>D<sub>6d</sub></i> are actually forbidden because they contain <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotations</a> with n=8 and 12 respectively. The 27 point groups in the table plus <i>T</i>, <i>T<sub>d</sub></i>, <i>T<sub>h</sub></i>, <i>O</i> and <i>O<sub>h</sub></i> constitute 32 crystallographic point groups. </p> <div class="mw-heading mw-heading3"><h3 id="Hermann–Mauguin_notation"><span id="Hermann.E2.80.93Mauguin_notation"></span>Hermann–Mauguin notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=3" title="Edit section: Hermann–Mauguin notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Hermann–Mauguin notation</a></div> <p>An abbreviated form of the <a href="/wiki/Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Hermann–Mauguin notation</a> commonly used for <a href="/wiki/Space_group" title="Space group">space groups</a> also serves to describe crystallographic point groups. Group names are </p> <table class="wikitable"> <tbody><tr> <th>Crystal family </th> <th>Crystal system </th> <th colspan="7">Group names </th></tr> <tr> <th colspan="2"><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">Cubic</a> </th> <td>23</td> <td>m<span style="text-decoration:overline;">3</span></td> <td></td> <td>432</td> <td><span style="text-decoration:overline;">4</span>3m</td> <td>m<span style="text-decoration:overline;">3</span>m</td> <td> </td></tr> <tr> <th rowspan="2"><a href="/wiki/Hexagonal_crystal_family" title="Hexagonal crystal family">Hexagonal</a> </th> <th>Hexagonal </th> <td>6</td> <td><span style="text-decoration:overline;">6</span></td> <td><sup>6</sup>⁄<sub>m</sub></td> <td>622</td> <td>6mm</td> <td><span style="text-decoration:overline;">6</span>m2</td> <td>6/mmm </td></tr> <tr> <th>Trigonal </th> <td>3</td> <td><span style="text-decoration:overline;">3</span></td> <td></td> <td>32</td> <td>3m</td> <td><span style="text-decoration:overline;">3</span>m</td> <td> </td></tr> <tr> <th colspan="2"><a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">Tetragonal</a> </th> <td>4</td> <td><span style="text-decoration:overline;">4</span></td> <td><sup>4</sup>⁄<sub>m</sub></td> <td>422</td> <td>4mm</td> <td><span style="text-decoration:overline;">4</span>2m</td> <td>4/mmm </td></tr> <tr> <th colspan="2"><a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">Orthorhombic</a> </th> <td></td> <td></td> <td></td> <td>222</td> <td></td> <td>mm2</td> <td>mmm </td></tr> <tr> <th colspan="2"><a href="/wiki/Monoclinic_crystal_system" title="Monoclinic crystal system">Monoclinic</a> </th> <td>2</td> <td></td> <td><sup>2</sup>⁄<sub>m</sub></td> <td></td> <td>m</td> <td></td> <td> </td></tr> <tr> <th colspan="2"><a href="/wiki/Triclinic_crystal_system" title="Triclinic crystal system">Triclinic</a> </th> <td>1</td> <td><span style="text-decoration:overline;">1</span></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="The_correspondence_between_different_notations">The correspondence between different notations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=4" title="Edit section: The correspondence between different notations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th rowspan="2">Crystal family </th> <th rowspan="2"><a href="/wiki/Crystal_system" title="Crystal system">Crystal system</a> </th> <th colspan="2"><a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Hermann-Mauguin</a> </th> <th rowspan="2">Shubnikov<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </th> <th rowspan="2"><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoenflies</a> </th> <th rowspan="2"><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orbifold</a> </th> <th rowspan="2"><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter</a> </th> <th rowspan="2">Order </th></tr> <tr align="center"> <th>(full) </th> <th>(short) </th></tr> <tr align="center"> <th rowspan="2" colspan="2"><a href="/wiki/Triclinic_crystal_system" title="Triclinic crystal system">Triclinic</a> </th> <td>1</td> <td>1</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98c9e64d6aa790731df88a02cc0d018cce78b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 1\ }"></span></td> <td><i>C<sub>1</sub></i></td> <td>11</td> <td>[ ]<sup>+</sup></td> <td>1 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">1</span></td> <td><span style="text-decoration:overline;">1</span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d43d94d26b8440864ad05ab9ec1e0f887c0b8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\tilde {2}}}"></span></td> <td><i>C<sub>i</sub> = S<sub>2</sub></i></td> <td>×</td> <td>[2<sup>+</sup>,2<sup>+</sup>]</td> <td>2 </td></tr> <tr align="center"> <th rowspan="3" colspan="2"><a href="/wiki/Monoclinic_crystal_system" title="Monoclinic crystal system">Monoclinic</a> </th> <td>2</td> <td>2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109498ca78d5a57ae4967b4f25d9b6e986a58528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 2\ }"></span></td> <td><i>C<sub>2</sub></i></td> <td>22</td> <td>[2]<sup>+</sup></td> <td>2 </td></tr> <tr align="center"> <td>m</td> <td>m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f753d46b8449abfe4aab6f5f1058188e46492f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.621ex; height:1.676ex;" alt="{\displaystyle m\ }"></span></td> <td><i>C<sub>s</sub> = C<sub>1h</sub></i></td> <td>*</td> <td>[ ]</td> <td>2 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ced983c01568491f53af2982d59fbff053092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{m}}}"></span></td> <td>2/m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3a12cf01b2f84549b5458d21e72c71465094af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle 2:m\ }"></span></td> <td><i>C<sub>2h</sub></i></td> <td>2*</td> <td>[2,2<sup>+</sup>]</td> <td>4 </td></tr> <tr align="center"> <th rowspan="3" colspan="2"><a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">Orthorhombic</a> </th> <td>222</td> <td>222</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2:2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>:</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2:2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742f1db191cb8c146865c1f5f2010e02c0233fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:2.176ex;" alt="{\displaystyle 2:2\ }"></span></td> <td><i>D<sub>2</sub> = V</i></td> <td>222</td> <td>[2,2]<sup>+</sup></td> <td>4 </td></tr> <tr align="center"> <td>mm2</td> <td>mm2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2b84ebf0a3699f511547ccb211cb8b207649b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle 2\cdot m\ }"></span></td> <td><i>C<sub>2v</sub></i></td> <td>*22</td> <td>[2]</td> <td>4 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d955ae7c54db2cc8ee3829103f444ade2552824b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.837ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"></span></td> <td>mmm</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot 2:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <mn>2</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot 2:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004e1661af14b3ff25303ca1eacaa0bfc1b9f148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.44ex; height:2.176ex;" alt="{\displaystyle m\cdot 2:m\ }"></span></td> <td><i>D<sub>2h</sub></i> = <i>V<sub>h</sub></i></td> <td>*222</td> <td>[2,2]</td> <td>8 </td></tr> <tr align="center"> <th rowspan="7" colspan="2"><a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">Tetragonal</a> </th> <td>4</td> <td>4</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f313d09bc2ad8b592f1e7a19bd7cb0fe64565efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 4\ }"></span></td> <td><i>C<sub>4</sub></i></td> <td>44</td> <td>[4]<sup>+</sup></td> <td>4 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span></td> <td><span style="text-decoration:overline;">4</span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>4</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/528cb1c9ad6c97341e2d2e064cf441ca22279f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\tilde {4}}}"></span> </td> <td><i>S<sub>4</sub></i></td> <td>2×</td> <td>[2<sup>+</sup>,4<sup>+</sup>]</td> <td>4 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c594b208ea7ba95f8d6eece9dd9a81f476811c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {4}{m}}}"></span></td> <td>4/m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4396edf37c081dec11c40528840a9e93141e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle 4:m\ }"></span></td> <td><i>C<sub>4h</sub></i></td> <td>4*</td> <td>[2,4<sup>+</sup>]</td> <td>8 </td></tr> <tr align="center"> <td>422</td> <td>422</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4:2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>:</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4:2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a0d8b4bea2a7af614278ab17d75a9b5905f7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:2.176ex;" alt="{\displaystyle 4:2\ }"></span></td> <td><i>D<sub>4</sub></i></td> <td>422</td> <td>[4,2]<sup>+</sup></td> <td>8 </td></tr> <tr align="center"> <td>4mm</td> <td>4mm</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cc616cad61e60d9fff6eaf869d40aa666157ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle 4\cdot m\ }"></span></td> <td><i>C<sub>4v</sub></i></td> <td>*44</td> <td>[4]</td> <td>8 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span>2m</td> <td><span style="text-decoration:overline;">4</span>2m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {4}}\cdot m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>4</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {4}}\cdot m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4c8316d4f1241d6a18b0a4e4f0e7c812b05986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.882ex; height:2.676ex;" alt="{\displaystyle {\tilde {4}}\cdot m}"></span></td> <td><i>D<sub>2d</sub></i> = <i>V<sub>d</sub></i></td> <td>2*2</td> <td>[2<sup>+</sup>,4]</td> <td>8 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec8b5c1ca032047a4ac1d108525e3deef4c2927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.837ex; height:3.343ex;" alt="{\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"></span></td> <td>4/mmm</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot 4:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <mn>4</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot 4:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dc4f02a6f75bd94611ee219e86905f982e69e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.44ex; height:2.176ex;" alt="{\displaystyle m\cdot 4:m\ }"></span></td> <td><i>D<sub>4h</sub></i></td> <td>*422</td> <td>[4,2]</td> <td>16 </td></tr> <tr align="center"> <th rowspan="12"><a href="/wiki/Hexagonal_crystal_family" title="Hexagonal crystal family">Hexagonal</a> </th> <th rowspan="5">Trigonal </th> <td>3</td> <td>3</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291adf564ec953470f6e085a47913556c596f264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 3\ }"></span></td> <td><i>C<sub>3</sub></i></td> <td>33</td> <td>[3]<sup>+</sup></td> <td>3 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">3</span></td> <td><span style="text-decoration:overline;">3</span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a6cf3f545337ea158e24e81a75df91462b5003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\tilde {6}}}"></span></td> <td><i>C<sub>3i</sub> = S<sub>6</sub></i></td> <td>3×</td> <td>[2<sup>+</sup>,6<sup>+</sup>]</td> <td>6 </td></tr> <tr align="center"> <td>32</td> <td>32</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3:2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>:</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3:2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/044f84c3373dbe4f600de6205a1a5d2844c0fe8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:2.176ex;" alt="{\displaystyle 3:2\ }"></span></td> <td><i>D<sub>3</sub></i></td> <td>322</td> <td>[3,2]<sup>+</sup></td> <td>6 </td></tr> <tr align="center"> <td>3m</td> <td>3m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ddaa0fb5fe4a597cb6cf1a0eeef1ad018a1dfd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle 3\cdot m\ }"></span></td> <td><i>C<sub>3v</sub></i></td> <td>*33</td> <td>[3]</td> <td>6 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">3</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ced983c01568491f53af2982d59fbff053092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{m}}}"></span></td> <td><span style="text-decoration:overline;">3</span>m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {6}}\cdot m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {6}}\cdot m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91dae018e90d7e625a76128aa9289e720111e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.882ex; height:2.676ex;" alt="{\displaystyle {\tilde {6}}\cdot m}"></span></td> <td><i>D<sub>3d</sub></i></td> <td>2*3</td> <td>[2<sup>+</sup>,6]</td> <td>12 </td></tr> <tr align="center"> <th rowspan="7">Hexagonal </th> <td>6</td> <td>6</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b62e16f30d22a094402fb958a7080144f6aa10c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 6\ }"></span></td> <td><i>C<sub>6</sub></i></td> <td>66</td> <td>[6]<sup>+</sup></td> <td>6 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">6</span></td> <td><span style="text-decoration:overline;">6</span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90088bdee66f6dd1a50fc958f32816a7fb944cf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle 3:m\ }"></span></td> <td><i>C<sub>3h</sub></i></td> <td>3*</td> <td>[2,3<sup>+</sup>]</td> <td>6 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {6}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>6</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {6}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783ebdf4ab903d3669591d3243c51093ce3bff13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {6}{m}}}"></span></td> <td>6/m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46438eea665eab0658e3a1032a1b463e939c36d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle 6:m\ }"></span></td> <td><i>C<sub>6h</sub></i></td> <td>6*</td> <td>[2,6<sup>+</sup>]</td> <td>12 </td></tr> <tr align="center"> <td>622</td> <td>622</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6:2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>:</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6:2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1c6e5b2372afbdb1437c3add6143fa3abf06f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.843ex; height:2.176ex;" alt="{\displaystyle 6:2\ }"></span></td> <td><i>D<sub>6</sub></i></td> <td>622</td> <td>[6,2]<sup>+</sup></td> <td>12 </td></tr> <tr align="center"> <td>6mm</td> <td>6mm</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\cdot m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>⋅</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\cdot m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67abeed8267261dc298c9ea7d5c094876d43020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle 6\cdot m\ }"></span></td> <td><i>C<sub>6v</sub></i></td> <td>*66</td> <td>[6]</td> <td>12 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">6</span>m2</td> <td><span style="text-decoration:overline;">6</span>m2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot 3:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <mn>3</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot 3:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09fd0f81cab6360594116a898c8a92c9546e1a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.44ex; height:2.176ex;" alt="{\displaystyle m\cdot 3:m\ }"></span></td> <td><i>D<sub>3h</sub></i></td> <td>*322</td> <td>[3,2]</td> <td>12 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>6</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34761ec14c764078c0c432c0842bba500ffe59fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.837ex; height:3.343ex;" alt="{\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}"></span></td> <td>6/mmm</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot 6:m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <mn>6</mn> <mo>:</mo> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot 6:m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d251c8c432c8c021e41154e8e65092c5d364472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.44ex; height:2.176ex;" alt="{\displaystyle m\cdot 6:m\ }"></span></td> <td><i>D<sub>6h</sub></i></td> <td>*622</td> <td>[6,2]</td> <td>24 </td></tr> <tr align="center"> <th rowspan="5" colspan="2"><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">Cubic</a> </th> <td>23</td> <td>23</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3/2\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3/2\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8860b88d493d7f92125f7dc653681ba2187d2ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 3/2\ }"></span></td> <td><i>T</i></td> <td>332</td> <td>[3,3]<sup>+</sup></td> <td>12 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ced983c01568491f53af2982d59fbff053092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{m}}}"></span><span style="text-decoration:overline;">3</span></td> <td>m<span style="text-decoration:overline;">3</span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {6}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {6}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91da8d5274e27595c60f002b8f7c8c8b0749b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:3.176ex;" alt="{\displaystyle {\tilde {6}}/2}"></span></td> <td><i>T<sub>h</sub></i></td> <td>3*2</td> <td>[3<sup>+</sup>,4]</td> <td>24 </td></tr> <tr align="center"> <td>432</td> <td>432</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3/4\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3/4\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e77531ca3bbc7d89cff24ff3800b27b3d2f52ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle 3/4\ }"></span></td> <td><i>O</i></td> <td>432</td> <td>[4,3]<sup>+</sup></td> <td>24 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span>3m</td> <td><span style="text-decoration:overline;">4</span>3m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3/{\tilde {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>4</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3/{\tilde {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b595534794430094a973b8428ced7a09fc456f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:3.176ex;" alt="{\displaystyle 3/{\tilde {4}}}"></span></td> <td><i>T<sub>d</sub></i></td> <td>*332</td> <td>[3,3]</td> <td>24 </td></tr> <tr align="center"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c594b208ea7ba95f8d6eece9dd9a81f476811c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {4}{m}}}"></span><span style="text-decoration:overline;">3</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ced983c01568491f53af2982d59fbff053092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.279ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{m}}}"></span></td> <td>m<span style="text-decoration:overline;">3</span>m</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {6}}/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {6}}/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2877a9393558921f3a4e2500f124579fafc6fdbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:3.176ex;" alt="{\displaystyle {\tilde {6}}/4}"></span></td> <td><i>O<sub>h</sub></i></td> <td>*432</td> <td>[4,3]</td> <td>48 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Isomorphisms">Isomorphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=5" title="Edit section: Isomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Crystal_structure#Crystal_systems" title="Crystal structure">Crystal structure § Crystal systems</a></div> <p>Many of the crystallographic point groups share the same internal structure. For example, the point groups <span style="text-decoration:overline;">1</span>, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> C<sub>2</sub>. All <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> groups are of the same <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a>, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th><a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Hermann-Mauguin</a> </th> <th><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schoenflies</a> </th> <th><a href="/wiki/Order_(group_theory)" title="Order (group theory)">Order</a> </th> <th colspan="2"><a href="/wiki/List_of_small_groups" title="List of small groups">Abstract group</a> </th></tr> <tr align="center"> <td>1</td> <td><i>C<sub>1</sub></i></td> <td>1</td> <td><a href="/wiki/Trivial_group" title="Trivial group">C<sub>1</sub></a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d25e00207b473def7c961593ce3cd98dd80a49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{1}^{1}}"></span> </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">1</span></td> <td><i>C<sub>i</sub> = S<sub>2</sub></i></td> <td>2</td> <td rowspan="3"><a href="/wiki/Cyclic_group" title="Cyclic group">C<sub>2</sub></a></td> <td rowspan="3"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1174a2e4bde969cc5660ac37e7cf2d449f34c799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{2}^{1}}"></span> </td></tr> <tr align="center"> <td>2</td> <td><i>C<sub>2</sub></i></td> <td>2 </td></tr> <tr align="center"> <td>m</td> <td><i>C<sub>s</sub> = C<sub>1h</sub></i></td> <td>2 </td></tr> <tr align="center"> <td>3</td> <td><i>C<sub>3</sub></i></td> <td>3</td> <td><a href="/wiki/Cyclic_group" title="Cyclic group">C<sub>3</sub></a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{3}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{3}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7dc3dc80ca387d2b2f77d680491e28635fda9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{3}^{1}}"></span> </td></tr> <tr align="center"> <td>4</td> <td><i>C<sub>4</sub></i></td> <td>4</td> <td rowspan="2"><a href="/wiki/Cyclic_group" title="Cyclic group">C<sub>4</sub></a></td> <td rowspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{4}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{4}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e4372d852751c4b2b539bac2478e5989bb895f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{4}^{1}}"></span> </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span></td> <td><i>S<sub>4</sub></i></td> <td>4 </td></tr> <tr align="center"> <td>2/m</td> <td> <i>C<sub>2h</sub></i></td> <td>4</td> <td rowspan="3"><a href="/wiki/Klein_four-group" title="Klein four-group">D<sub>2</sub></a> = C<sub>2</sub> × C<sub>2</sub></td> <td rowspan="3"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{4}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{4}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64b5a38e50035b536bdab5baaf96bf595aa0ca44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{4}^{2}}"></span> </td></tr> <tr align="center"> <td> 222</td> <td><i>D<sub>2</sub> = V</i></td> <td>4 </td></tr> <tr align="center"> <td>mm2</td> <td><i>C<sub>2v</sub></i></td> <td> 4 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">3</span></td> <td><i>C<sub>3i</sub> = S<sub>6</sub></i></td> <td>6</td> <td rowspan="3"><a href="/wiki/Cyclic_group" title="Cyclic group">C<sub>6</sub></a></td> <td rowspan="3"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{6}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{6}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1c517ac3c9f730675e711487dca38f6fd5df30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{6}^{1}}"></span> </td></tr> <tr align="center"> <td>6</td> <td><i>C<sub>6</sub></i></td> <td>6 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">6</span></td> <td><i>C<sub>3h</sub></i></td> <td>6 </td></tr> <tr align="center"> <td>32</td> <td><i>D<sub>3</sub></i></td> <td>6</td> <td rowspan="2"><a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">D<sub>3</sub></a></td> <td rowspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{6}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{6}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b6123d14a431a3408d288a8ab33091bbb88be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{6}^{2}}"></span> </td></tr> <tr align="center"> <td>3m</td> <td><i>C<sub>3v</sub></i></td> <td>6 </td></tr> <tr align="center"> <td>mmm</td> <td><i>D<sub>2h</sub></i> = <i>V<sub>h</sub></i></td> <td>8</td> <td>D<sub>2</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{8}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{8}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/308c156140d85eda4c42eab954996bb4a09670aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{8}^{3}}"></span> </td></tr> <tr align="center"> <td> 4/m</td> <td><i>C<sub>4h</sub></i></td> <td>8</td> <td>C<sub>4</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{8}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{8}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ada708821539b862c5b8ba0ca3cff6fa41dfee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{8}^{2}}"></span> </td></tr> <tr align="center"> <td>422</td> <td><i>D<sub>4</sub></i></td> <td>8</td> <td rowspan="3"><a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">D<sub>4</sub></a></td> <td rowspan="3"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{8}^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{8}^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fc94ae5b040781d41290e6d336c6c184456947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.881ex; height:3.176ex;" alt="{\displaystyle G_{8}^{4}}"></span> </td></tr> <tr align="center"> <td>4mm</td> <td><i>C<sub>4v</sub></i></td> <td>8 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span>2m</td> <td><i>D<sub>2d</sub></i> = <i>V<sub>d</sub></i></td> <td>8 </td></tr> <tr align="center"> <td>6/m</td> <td><i>C<sub>6h</sub></i></td> <td>12</td> <td>C<sub>6</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{12}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{12}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8079425dfa62db8c1807dd37d2e9f0ca33b50832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{12}^{2}}"></span> </td></tr> <tr align="center"> <td>23</td> <td><i>T</i></td> <td>12</td> <td><a href="/wiki/Alternating_group" title="Alternating group">A<sub>4</sub></a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{12}^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{12}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f72de8565765aa433e8b9545a2568320c557c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{12}^{5}}"></span> </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">3</span>m</td> <td><i>D<sub>3d</sub></i></td> <td>12</td> <td rowspan="4"><a href="/wiki/Dihedral_group" title="Dihedral group">D<sub>6</sub></a></td> <td rowspan="4"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{12}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{12}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ca0f72df0a61c70ca8cd7397c627104a690b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{12}^{3}}"></span> </td></tr> <tr align="center"> <td>622</td> <td><i>D<sub>6</sub></i></td> <td>12 </td></tr> <tr align="center"> <td>6mm</td> <td><i>C<sub>6v</sub></i></td> <td>12 </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">6</span>m2</td> <td><i>D<sub>3h</sub></i></td> <td>12 </td></tr> <tr align="center"> <td>4/mmm</td> <td><i>D<sub>4h</sub></i></td> <td>16</td> <td>D<sub>4</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{16}^{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{16}^{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e488c8251fb28ce7cc1cbb2aad22e312dcb7b4af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{16}^{9}}"></span> </td></tr> <tr align="center"> <td>6/mmm</td> <td><i>D<sub>6h</sub></i></td> <td>24</td> <td>D<sub>6</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{24}^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{24}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f2d88a193181d4507b9e444282e377c8423046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{24}^{5}}"></span> </td></tr> <tr align="center"> <td>m<span style="text-decoration:overline;">3</span></td> <td><i>T<sub>h</sub></i></td> <td>24</td> <td>A<sub>4</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{24}^{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{24}^{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87c5da95e094a54481b431de7a060a671010453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{24}^{10}}"></span> </td></tr> <tr align="center"> <td>432</td> <td><i>O</i>  </td> <td>24</td> <td rowspan="2"><a href="/wiki/Symmetric_group" title="Symmetric group">S<sub>4</sub></a></td> <td rowspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{24}^{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{24}^{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deda8b880ce0e1f24bcdfa9186169e117859585d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{24}^{7}}"></span> </td></tr> <tr align="center"> <td><span style="text-decoration:overline;">4</span>3m</td> <td><i>T<sub>d</sub></i></td> <td>24 </td></tr> <tr align="center"> <td>m<span style="text-decoration:overline;">3</span>m</td> <td><i>O<sub>h</sub></i></td> <td>48</td> <td>S<sub>4</sub> × C<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{48}^{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>48</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{48}^{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2001552d2fb44e9c1dc1e71e46328ebf37028c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.703ex; height:3.176ex;" alt="{\displaystyle G_{48}^{7}}"></span> </td></tr></tbody></table> <p>This table makes use of <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, C<sub>4</sub>, C<sub>6</sub>), <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral groups</a> (D<sub>2</sub>, D<sub>3</sub>, D<sub>4</sub>, D<sub>6</sub>), one of the <a href="/wiki/Alternating_group" title="Alternating group">alternating groups</a> (A<sub>4</sub>), and one of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric groups</a> (S<sub>4</sub>). Here the symbol " × " indicates a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Deriving_the_crystallographic_point_group_(crystal_class)_from_the_space_group"><span id="Deriving_the_crystallographic_point_group_.28crystal_class.29_from_the_space_group"></span>Deriving the crystallographic point group (crystal class) from the space group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=6" title="Edit section: Deriving the crystallographic point group (crystal class) from the space group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Leave out the <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a> type.</li> <li>Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.)</li> <li>Axes of rotation, <a href="/wiki/Improper_rotation" title="Improper rotation">rotoinversion</a> axes, and mirror planes remain unchanged.</li></ol> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Molecular_symmetry" title="Molecular symmetry">Molecular symmetry</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Space_group" title="Space group">Space group</a></li> <li><a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">Point groups in three dimensions</a></li> <li><a href="/wiki/Crystal_system" title="Crystal system">Crystal system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.today/20130704042455/http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/">"(International Tables) Abstract"</a>. Archived from <a rel="nofollow" class="external text" href="http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/">the original</a> on 2013-07-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%28International+Tables%29+Abstract&rft_id=http%3A%2F%2Fit.iucr.org%2FAb%2Fch12o1v0001%2Fsec12o1o3%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystallographic+point+group" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNovak1995" class="citation journal cs1">Novak, I (1995-07-18). "Molecular isomorphism". <i>European Journal of Physics</i>. <b>16</b> (4). IOP Publishing: 151–153. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995EJPh...16..151N">1995EJPh...16..151N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F16%2F4%2F001">10.1088/0143-0807/16/4/001</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0143-0807">0143-0807</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:250887121">250887121</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=Molecular+isomorphism&rft.volume=16&rft.issue=4&rft.pages=151-153&rft.date=1995-07-18&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F16%2F4%2F001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250887121%23id-name%3DS2CID&rft.issn=0143-0807&rft_id=info%3Abibcode%2F1995EJPh...16..151N&rft.aulast=Novak&rft.aufirst=I&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystallographic+point+group" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystallographic_point_group&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://archive.today/20130704033345/http://it.iucr.org/Ab/ch12o1v0001/">Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820</a></li> <li><a rel="nofollow" class="external text" href="https://archive.today/20130704032551/http://it.iucr.org/Ab/ch10o1v0001/table10o1o2o4/">Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120204104121/http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html">Pictorial overview of the 32 groups</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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.navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Crystal_systems" title="Template:Crystal systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Crystal_systems&action=edit&redlink=1" class="new" title="Template talk:Crystal systems (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Crystal_systems" title="Special:EditPage/Template:Crystal systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Crystal_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Crystal_system" title="Crystal system">Crystal systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li> <li><a class="mw-selflink selflink">Crystallographic point group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Seven 3D systems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triclinic_crystal_system" title="Triclinic crystal system">triclinic</a> (anorthic) <span typeof="mw:File"><a href="/wiki/File:Triclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/30px-Triclinic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/45px-Triclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/60px-Triclinic.svg.png 2x" data-file-width="129" data-file-height="149" /></a></span></li> <li><a href="/wiki/Monoclinic_crystal_system" title="Monoclinic crystal system">monoclinic</a> <span typeof="mw:File"><a href="/wiki/File:Monoclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/30px-Monoclinic.svg.png" decoding="async" width="30" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/45px-Monoclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/60px-Monoclinic.svg.png 2x" data-file-width="114" data-file-height="143" /></a></span></li> <li><a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">orthorhombic</a> <span typeof="mw:File"><a href="/wiki/File:Orthorhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/30px-Orthorhombic.svg.png" decoding="async" width="30" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/45px-Orthorhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/60px-Orthorhombic.svg.png 2x" data-file-width="108" data-file-height="142" /></a></span></li> <li><a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">tetragonal</a> <span typeof="mw:File"><a href="/wiki/File:Tetragonal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/30px-Tetragonal.svg.png" decoding="async" width="30" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/45px-Tetragonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/60px-Tetragonal.svg.png 2x" data-file-width="108" data-file-height="165" /></a></span></li> <li><a href="/wiki/Hexagonal_crystal_family" title="Hexagonal crystal family">trigonal & hexagonal</a> <span typeof="mw:File"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/30px-Rhombohedral.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/45px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/60px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span><span typeof="mw:File"><a href="/wiki/File:Hexagonal_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/30px-Hexagonal_lattice.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/45px-Hexagonal_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/60px-Hexagonal_lattice.svg.png 2x" data-file-width="160" data-file-height="186" /></a></span></li> <li><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic</a> (isometric) <span typeof="mw:File"><a href="/wiki/File:Cubic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/30px-Cubic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/45px-Cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/60px-Cubic.svg.png 2x" data-file-width="109" data-file-height="127" /></a></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Four 2D systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Oblique_lattice" title="Oblique lattice">oblique</a> <span typeof="mw:File"><a href="/wiki/File:2d_mp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/30px-2d_mp.svg.png" decoding="async" width="30" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/45px-2d_mp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/60px-2d_mp.svg.png 2x" data-file-width="140" data-file-height="170" /></a></span></li> <li><a href="/wiki/Rectangular_lattice" title="Rectangular lattice">rectangular</a> <span typeof="mw:File"><a href="/wiki/File:2d_op_rectangular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/30px-2d_op_rectangular.svg.png" decoding="async" width="30" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/45px-2d_op_rectangular.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/60px-2d_op_rectangular.svg.png 2x" data-file-width="210" data-file-height="170" /></a></span></li> <li><a href="/wiki/Square_lattice" title="Square lattice">square</a> <span typeof="mw:File"><a href="/wiki/File:2d_tp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/30px-2d_tp.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/45px-2d_tp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/60px-2d_tp.svg.png 2x" data-file-width="170" data-file-height="170" /></a></span></li> <li><a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">hexagonal</a> <span typeof="mw:File"><a href="/wiki/File:2d_hp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/30px-2d_hp.svg.png" decoding="async" width="30" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/45px-2d_hp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/2d_hp.svg/60px-2d_hp.svg.png 2x" data-file-width="290" data-file-height="280" /></a></span></li></ul> </div></td></tr></tbody></table></div>    </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" 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