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href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-術語" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#術語"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>術語</span> </div> </a> <ul id="toc-術語-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-小非可換群的列表" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#小非可換群的列表"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>小非可換群的列表</span> </div> </a> <ul id="toc-小非可換群的列表-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-合併列表" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#合併列表"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>合併列表</span> </div> </a> <ul id="toc-合併列表-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-小群圖書館" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#小群圖書館"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>小群圖書館</span> </div> </a> <ul id="toc-小群圖書館-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-另見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#另見"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>另見</span> </div> </a> <ul id="toc-另見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">小群列表</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。14种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-14" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">14种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llista_de_grups_petits" title="Llista de grups petits – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Llista de grups petits" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Sm%C3%A5_grupper" title="Små grupper – 丹麦语" lang="da" hreflang="da" data-title="Små grupper" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17506997 badge-featuredlist mw-list-item" title="特色列表"><a href="https://de.wikipedia.org/wiki/Liste_kleiner_Gruppen" title="Liste kleiner Gruppen – 德语" lang="de" hreflang="de" data-title="Liste kleiner Gruppen" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/List_of_small_groups" title="List of small groups – 英语" lang="en" hreflang="en" data-title="List of small groups" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anexo:Grupos_finitos_de_orden_bajo" title="Anexo:Grupos finitos de orden bajo – 西班牙语" lang="es" hreflang="es" data-title="Anexo:Grupos finitos de orden bajo" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Liste_des_petits_groupes" title="Liste des petits groupes – 法语" lang="fr" hreflang="fr" data-title="Liste des petits groupes" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kis_elemsz%C3%A1m%C3%BA_v%C3%A9ges_csoportok_list%C3%A1ja" title="Kis elemszámú véges csoportok listája – 匈牙利语" lang="hu" hreflang="hu" data-title="Kis elemszámú véges csoportok listája" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Tabella_de_parve_gruppos" title="Tabella de parve gruppos – 国际语" lang="ia" hreflang="ia" data-title="Tabella de parve gruppos" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Daftar_grup_kecil" title="Daftar grup kecil – 印度尼西亚语" lang="id" hreflang="id" data-title="Daftar grup kecil" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tavola_dei_gruppi_piccoli" title="Tavola dei gruppi piccoli – 意大利语" lang="it" hreflang="it" data-title="Tavola dei gruppi piccoli" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%91%EC%9D%80_%EA%B5%B0%EC%9D%98_%EB%AA%A9%EB%A1%9D" title="작은 군의 목록 – 韩语" lang="ko" hreflang="ko" data-title="작은 군의 목록" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lijst_van_kleine_groepen" title="Lijst van kleine groepen – 荷兰语" lang="nl" hreflang="nl" data-title="Lijst van kleine groepen" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%B3%D1%80%D1%83%D0%BF%D0%BF_%D0%BC%D0%B0%D0%BB%D0%BE%D0%B3%D0%BE_%D0%BF%D0%BE%D1%80%D1%8F%D0%B4%D0%BA%D0%B0" title="Список групп малого порядка – 俄语" lang="ru" hreflang="ru" data-title="Список групп малого порядка" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Seznam_malih_grup" title="Seznam malih grup – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Seznam malih grup" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1332566#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&redlink=1" rel="discussion" class="new" title="关于此页面的讨论（页面不存在）[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a 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</div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <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="zh" dir="ltr"><p>下面的<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>列表包含著以<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%A7%8B" title="群同構">群同構</a>來分之小<a href="/wiki/%E9%9A%8E_(%E7%BE%A4%E8%AB%96)" title="階 (群論)">階</a><a href="/wiki/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群">有限群</a>。 </p><p>這個列表可以被用來決定一個給定的有限群<i>G</i>會同構於哪一種群：首先確定<i>G</i>的階，然後再找下面列表中有相同階的候選群。若知道<i>G</i>為可換與否，某些的候選群便可以立刻被刪掉。為了分別剩下的候選群，可以看給定之群內每個元素的階，並對照候選群內每個元素的階。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="術語"><span id=".E8.A1.93.E8.AA.9E"></span>術語</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=1" title="编辑章节：術語"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Z<sub><i>n</i></sub>：其階為<i>n</i>之<a href="/wiki/%E5%BE%AA%E7%92%B0%E7%BE%A4" title="循環群">循環群</a>(通常<i>C</i><sub><i>n</i></sub>或Z/<i>n</i>Z之符號也會被使用)。</li> <li>Dih<sub><i>n</i></sub>：其階為2<i>n</i>的<a href="/wiki/%E4%BA%8C%E9%9D%A2%E9%AB%94%E7%BE%A4" title="二面體群">二面體群</a>(通常<i>D</i><sub><i>n</i></sub>之符號也會被使用，有時則會用<i>D</i><sub>2<i>n</i></sub>)。</li> <li><i>S</i><sub><i>n</i></sub>：<i>n</i>階的<a href="/wiki/%E5%AF%B9%E7%A7%B0%E7%BE%A4_(n%E6%AC%A1%E5%AF%B9%E7%A7%B0%E7%BE%A4)" title="对称群 (n次对称群)">對稱群</a>，包含有<a href="/wiki/%E9%9A%8E%E4%B9%98" title="階乘"><i>n</i>!</a>個<i>n</i>個元素的<a href="/wiki/%E7%BD%AE%E6%8F%9B" class="mw-redirect" title="置換">置換</a>。</li> <li><i>A</i><sub><i>n</i></sub>：<i>n</i>階的<a href="/wiki/%E4%BA%A4%E9%94%99%E7%BE%A4" title="交错群">交錯群</a>，包含有<i>n</i>!/2個<i>n</i>個元素的<a href="/wiki/%E5%81%B6%E7%BD%AE%E6%8D%A2" class="mw-redirect" title="偶置换">偶置換</a>。</li> <li>Dic<sub><i>n</i></sub>：其階為4<i>n</i>的<a href="/w/index.php?title=%E9%9B%99%E5%BE%AA%E7%92%B0%E7%BE%A4&action=edit&redlink=1" class="new" title="雙循環群（页面不存在）">雙循環群</a>。</li></ul> <p>Z<sub><i>n</i></sub>和Dih<sub><i>n</i></sub>之符號在<a href="/wiki/%E4%B8%89%E7%B6%AD%E9%BB%9E%E7%BE%A4" title="三維點群">三維點群</a><i>C</i><sub><i>n</i></sub>和<i>D</i><sub><i>n</i></sub>中有著沒有相同符號的優點。其存在著多於此兩類的<a href="/w/index.php?title=%E7%AD%89%E8%B7%9D%E5%90%8C%E6%A7%8B%E7%BE%A4&action=edit&redlink=1" class="new" title="等距同構群（页面不存在）">等距同構群</a>，但這些都有著相同的抽象群類型。 </p><p>符號<i>G</i> × <i>H</i>表示是兩個群的<a href="/wiki/%E7%9B%B4%E7%A7%AF" title="直积">直積</a>。<a href="/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%BE%A4" class="mw-redirect" title="阿貝爾群">阿貝爾群</a>和<a href="/wiki/%E7%B0%A1%E5%96%AE%E7%BE%A4" class="mw-redirect" title="簡單群">簡單群</a>會加上注釋(對小於60階之群，簡單群會恰好是循環群Z<sub><i>n</i></sub>，其中的<i>n</i>為質數。)下面會以等號(=)來標註同構。 </p><p><a href="/wiki/%E7%92%B0%E5%9C%96" title="環圖">環圖</a>內的單位元素會以黑圓圈來表示。圖環不能唯一地表示一個群之最小階為16。 </p><p>下面列表中的子群，當然群和群自身並不會被列出來。 </p> <div class="mw-heading mw-heading2"><h2 id="小非可換群的列表"><span id=".E5.B0.8F.E9.9D.9E.E5.8F.AF.E6.8F.9B.E7.BE.A4.E7.9A.84.E5.88.97.E8.A1.A8"></span>小非可換群的列表</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=2" title="编辑章节：小非可換群的列表"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>另見<a href="/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%BE%A4#小阿貝爾群列表" class="mw-redirect" title="阿貝爾群">小阿貝爾群列表</a>和下面合併的列表。 </p><p>注意如「3×Z<sub>2</sub>」之標記表示其有3個Z<sub>2</sub>類型的子群(而不是Z<sub>2</sub>的一個左陪集)，而其他地方裡的×則表示<a href="/wiki/%E7%9B%B4%E7%A7%AF" title="直积">直積</a>。 </p> <table class="wikitable"> <tbody><tr> <th>階</th> <th>群</th> <th>子群</th> <th>性質</th> <th>環圖</th> </tr> <tr> <th>6</th> <td><a href="/w/index.php?title=6%E9%9A%8E%E4%BA%8C%E9%9D%A2%E9%AB%94%E7%BE%A4&action=edit&redlink=1" class="new" title="6階二面體群（页面不存在）"><i>S</i><sub>3</sub> = Dih<sub>3</sub></a></td> <td>Z<sub>3</sub> , 3 × Z<sub>2</sub></td> <td>最小的非可換群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/32px-GroupDiagramMiniD6.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/48px-GroupDiagramMiniD6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/64px-GroupDiagramMiniD6.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">8</th> <td> <a href="/w/index.php?title=%E7%BE%A4%E7%9A%84%E4%BE%8B%E5%AD%90&action=edit&redlink=1" class="new" title="群的例子（页面不存在）">Dih<sub>4</sub></a></td> <td>Z<sub>4</sub>, 2 × Dih<sub>2</sub> , 5 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/32px-GroupDiagramMiniD8.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/48px-GroupDiagramMiniD8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/64px-GroupDiagramMiniD8.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/wiki/%E5%9B%9B%E5%85%83%E7%BE%A4" title="四元群">四元群</a>, <i>Q</i><sub>8</sub> = Dic<sub>2</sub> </td> <td> 3 × Z<sub>4</sub> , Z<sub>2</sub></td> <td>非可換；最小的<a href="/wiki/%E6%BC%A2%E5%BD%8C%E7%88%BE%E9%A0%93%E7%BE%A4" class="mw-redirect" title="漢彌爾頓群">漢彌爾頓群</a></td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQ8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/44px-GroupDiagramMiniQ8.svg.png" decoding="async" width="44" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/66px-GroupDiagramMiniQ8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/88px-GroupDiagramMiniQ8.svg.png 2x" data-file-width="44" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>10</th> <td>Dih<sub>5</sub></td> <td> Z<sub>5</sub> , 5 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/32px-GroupDiagramMiniD10.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/48px-GroupDiagramMiniD10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/64px-GroupDiagramMiniD10.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="3">12</th> <td> Dih<sub>6</sub> = Dih<sub>3</sub> × Z<sub>2</sub></td> <td>Z<sub>6</sub> , 2 × Dih<sub>3</sub> , 3 × Dih<sub>2</sub> , Z<sub>3</sub> , 7 × Z<sub>2</sub> </td><td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/64px-GroupDiagramMiniD12.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/96px-GroupDiagramMiniD12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/128px-GroupDiagramMiniD12.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> <i>A</i><sub>4</sub> </td> <td>Z<sub>2</sub><sup>2</sup>, 4 × Z<sub>3</sub>, 3 × Z<sub>2</sub> </td> <td>非可換；最小確定<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)" title="拉格朗日定理 (群論)">拉格朗日定理</a>之相反敘述不是對的群：沒有6階的子群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniA4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/64px-GroupDiagramMiniA4.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/96px-GroupDiagramMiniA4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/128px-GroupDiagramMiniA4.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dic<sub>3</sub> = Z<sub>3</sub>和Z<sub>4</sub>的<a href="/wiki/%E5%8D%8A%E7%9B%B4%E7%A7%AF" title="半直积">半直積</a>，其中Z<sub>4</sub>以反演作用於Z<sub>3</sub>上</td> <td>Z<sub>2</sub>, Z<sub>3</sub>, 3 × Z<sub>4</sub>, Z<sub>6</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniX12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/86px-GroupDiagramMiniX12.svg.png" decoding="async" width="86" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/129px-GroupDiagramMiniX12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/172px-GroupDiagramMiniX12.svg.png 2x" data-file-width="86" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="1">14</th> <td> Dih<sub>7</sub></td> <td> Z<sub>7</sub> , 7 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD14.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/64px-GroupDiagramMiniD14.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/96px-GroupDiagramMiniD14.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/128px-GroupDiagramMiniD14.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="9">16</th> <td>Dih<sub>8</sub></td> <td>Z<sub>8</sub> , 2 × Dih<sub>4</sub> , 4 × Dih<sub>2</sub> , Z<sub>4</sub> , 9 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/64px-GroupDiagramMiniD16.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/96px-GroupDiagramMiniD16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/128px-GroupDiagramMiniD16.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dih<sub>4</sub> × Z<sub>2</sub></td> <td> 2 × Dih<sub>4</sub> , Z<sub>4</sub> × Z<sub>2</sub> , 2 × Z<sub>2</sub><sup>3<sup>, 7 × Z<sub>2</sub><sup>2</sup> , 2 × Z<sub>4</sub> , 11 × Z<sub>2</sub> </sup></sup></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2D8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/76px-GroupDiagramMiniC2D8.svg.png" decoding="async" width="76" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/114px-GroupDiagramMiniC2D8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/152px-GroupDiagramMiniC2D8.svg.png 2x" data-file-width="76" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/wiki/%E5%9B%9B%E5%85%83%E7%BE%A4" title="四元群">廣義四元群</a>, <i>Q</i><sub>16</sub> = Dic<sub>4</sub></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQ16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/119px-GroupDiagramMiniQ16.svg.png" decoding="async" width="119" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/179px-GroupDiagramMiniQ16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/238px-GroupDiagramMiniQ16.svg.png 2x" data-file-width="119" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><i>Q</i><sub>8</sub> × Z<sub>2</sub></td> <td> </td> <td>非可換、<a href="/wiki/%E6%BC%A2%E5%BD%8C%E7%88%BE%E9%A0%93%E7%BE%A4" class="mw-redirect" title="漢彌爾頓群">漢彌爾頓群</a></td> <td><figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:GroupDiagramMiniC2Q8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/100px-GroupDiagramMiniC2Q8.svg.png" decoding="async" width="100" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/150px-GroupDiagramMiniC2Q8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/200px-GroupDiagramMiniC2Q8.svg.png 2x" data-file-width="100" data-file-height="64" /></a><figcaption><i>wrong</i></figcaption></figure></td> </tr> <tr> <td>16階之<a href="/w/index.php?title=%E6%93%AC%E4%BA%8C%E9%9D%A2%E9%AB%94%E7%BE%A4&action=edit&redlink=1" class="new" title="擬二面體群（页面不存在）">擬二面體群</a></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQH16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/84px-GroupDiagramMiniQH16.svg.png" decoding="async" width="84" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/126px-GroupDiagramMiniQH16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/168px-GroupDiagramMiniQH16.svg.png 2x" data-file-width="84" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>16階之<a href="/w/index.php?title=%E6%A8%A1%E7%BE%A4&action=edit&redlink=1" class="new" title="模群（页面不存在）">模群</a></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2C8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/64px-GroupDiagramMiniC2C8.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/96px-GroupDiagramMiniC2C8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/128px-GroupDiagramMiniC2C8.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>4</sub>和Z<sub>4</sub>的<a href="/w/index.php?title=%E5%8D%8A%E7%9B%B4%E5%92%8C&action=edit&redlink=1" class="new" title="半直和（页面不存在）">半直和</a>，其中一個以反演作用在另一個上</td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMinix3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/77px-GroupDiagramMinix3.svg.png" decoding="async" width="77" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/116px-GroupDiagramMinix3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/154px-GroupDiagramMinix3.svg.png 2x" data-file-width="77" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>由<a href="/wiki/%E6%B3%A1%E5%88%A9%E7%9F%A9%E9%99%A3" title="泡利矩陣">泡利矩陣</a>產生的群</td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2x2C4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/78px-GroupDiagramMiniC2x2C4.svg.png" decoding="async" width="78" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/117px-GroupDiagramMiniC2x2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/156px-GroupDiagramMiniC2x2C4.svg.png 2x" data-file-width="78" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><i>G</i><sub>4,4</sub></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniG44.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/72px-GroupDiagramMiniG44.svg.png" decoding="async" width="72" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/108px-GroupDiagramMiniG44.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/144px-GroupDiagramMiniG44.svg.png 2x" data-file-width="72" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="合併列表"><span id=".E5.90.88.E4.BD.B5.E5.88.97.E8.A1.A8"></span>合併列表</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=3" title="编辑章节：合併列表"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>階</th> <th>群</th> <th>子群</th> <th>性質</th> <th>環圖</th> </tr> <tr> <th>1</th> <td><a href="/wiki/%E5%B9%B3%E5%87%A1%E7%BE%A4" class="mw-redirect" title="平凡群">平凡群</a> = Z<sub>1</sub> = <i>S</i><sub>1</sub> = <i>A</i><sub>2</sub></td> <td>-</td> <td>平凡、循环、交错、对称、初等 </td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/32px-GroupDiagramMiniC1.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/48px-GroupDiagramMiniC1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/GroupDiagramMiniC1.svg/64px-GroupDiagramMiniC1.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>2</th> <td>Z<sub>2</sub> = <i>S</i><sub>2</sub> = Dih<sub>1</sub></td> <td>-</td> <td>可換、簡單、最小非當然群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/32px-GroupDiagramMiniC2.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/48px-GroupDiagramMiniC2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/GroupDiagramMiniC2.svg/64px-GroupDiagramMiniC2.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>3</th> <td>Z<sub>3</sub> = <i>A</i><sub>3</sub></td> <td>-</td> <td>可換、簡單</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/32px-GroupDiagramMiniC3.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/48px-GroupDiagramMiniC3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/GroupDiagramMiniC3.svg/64px-GroupDiagramMiniC3.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">4</th> <td>Z<sub>4</sub></td> <td>Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/32px-GroupDiagramMiniC4.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/48px-GroupDiagramMiniC4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/GroupDiagramMiniC4.svg/64px-GroupDiagramMiniC4.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E5%9B%9B%E5%85%83%E7%BE%A4" title="克莱因四元群">克萊因四元群</a> = Z<sub>2</sub> × Z<sub>2</sub> = Dih<sub>2</sub></td> <td>3 × Z<sub>2</sub></td> <td>可換、最小非循環群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/32px-GroupDiagramMiniD4.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/48px-GroupDiagramMiniD4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/GroupDiagramMiniD4.svg/64px-GroupDiagramMiniD4.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>5</th> <td>Z<sub>5</sub> </td> <td>-</td> <td>可換、簡單</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/32px-GroupDiagramMiniC5.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/48px-GroupDiagramMiniC5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/GroupDiagramMiniC5.svg/64px-GroupDiagramMiniC5.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">6</th> <td>Z<sub>6</sub> = Z<sub>2</sub> × Z<sub>3</sub></td> <td>Z<sub>2</sub> , Z<sub>3</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/32px-GroupDiagramMiniC6.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/48px-GroupDiagramMiniC6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/GroupDiagramMiniC6.svg/64px-GroupDiagramMiniC6.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/w/index.php?title=6%E9%9A%8E%E4%BA%8C%E9%9D%A2%E9%AB%94%E7%BE%A4&action=edit&redlink=1" class="new" title="6階二面體群（页面不存在）"><i>S</i><sub>3</sub> = Dih<sub>3</sub></a></td> <td>Z<sub>3</sub> , 3 × Z<sub>2</sub></td> <td>最小非可換群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/32px-GroupDiagramMiniD6.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/48px-GroupDiagramMiniD6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniD6.svg/64px-GroupDiagramMiniD6.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>7</th> <td>Z<sub>7</sub> </td> <td>-</td> <td>可換、簡單</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC7.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/32px-GroupDiagramMiniC7.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/48px-GroupDiagramMiniC7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/GroupDiagramMiniC7.svg/64px-GroupDiagramMiniC7.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="5">8</th> <td><a href="/wiki/%E5%AD%90%E7%BE%A4#例子" title="子群">Z<sub>8</sub></a></td> <td>Z<sub>4</sub> , Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/32px-GroupDiagramMiniC8.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/48px-GroupDiagramMiniC8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/GroupDiagramMiniC8.svg/64px-GroupDiagramMiniC8.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>2</sub> ×Z<sub>4</sub></td> <td>2 × Z<sub>4</sub> , 3 ×Z<sub>2</sub> , Dih<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2C4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GroupDiagramMiniC2C4.svg/32px-GroupDiagramMiniC2C4.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GroupDiagramMiniC2C4.svg/48px-GroupDiagramMiniC2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GroupDiagramMiniC2C4.svg/64px-GroupDiagramMiniC2C4.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> Z<sub>2</sub> × Z<sub>2</sub> × Z<sub>2</sub> = Dih<sub>2</sub> × Z<sub>2</sub></td> <td>7 × Z<sub>2</sub> × Z<sub>2</sub> , 7 × Z<sub>2</sub> </td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2x3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/32px-GroupDiagramMiniC2x3.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/48px-GroupDiagramMiniC2x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/GroupDiagramMiniC2x3.svg/64px-GroupDiagramMiniC2x3.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> <a href="/w/index.php?title=%E7%BE%A4%E7%9A%84%E4%BE%8B%E5%AD%90&action=edit&redlink=1" class="new" title="群的例子（页面不存在）">Dih<sub>4</sub></a></td> <td>Z<sub>4</sub>, 2 × Dih<sub>2</sub> , 5 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/32px-GroupDiagramMiniD8.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/48px-GroupDiagramMiniD8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniD8.svg/64px-GroupDiagramMiniD8.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/wiki/%E5%9B%9B%E5%85%83%E7%BE%A4" title="四元群">四元群</a>, <i>Q</i><sub>8</sub> = Dic<sub>2</sub> </td> <td> 3 × Z<sub>4</sub> , Z<sub>2</sub></td> <td>非可換、最小<a href="/wiki/%E6%BC%A2%E5%BD%8C%E7%88%BE%E9%A0%93%E7%BE%A4" class="mw-redirect" title="漢彌爾頓群">漢彌爾頓群</a></td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQ8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/44px-GroupDiagramMiniQ8.svg.png" decoding="async" width="44" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/66px-GroupDiagramMiniQ8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/GroupDiagramMiniQ8.svg/88px-GroupDiagramMiniQ8.svg.png 2x" data-file-width="44" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">9</th> <td>Z<sub>9</sub></td> <td> Z<sub>3</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC9.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/32px-GroupDiagramMiniC9.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/48px-GroupDiagramMiniC9.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GroupDiagramMiniC9.svg/64px-GroupDiagramMiniC9.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> Z<sub>3</sub> × Z<sub>3</sub></td> <td> 4 × Z<sub>3</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC3x2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/GroupDiagramMiniC3x2.svg/32px-GroupDiagramMiniC3x2.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/GroupDiagramMiniC3x2.svg/48px-GroupDiagramMiniC3x2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/GroupDiagramMiniC3x2.svg/64px-GroupDiagramMiniC3x2.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">10</th> <td>Z<sub>10</sub> = Z<sub>2</sub> × Z<sub>5</sub></td> <td> Z<sub>5</sub> , Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/32px-GroupDiagramMiniC10.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/48px-GroupDiagramMiniC10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/GroupDiagramMiniC10.svg/64px-GroupDiagramMiniC10.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dih<sub>5</sub></td> <td> Z<sub>5</sub> , 5 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/32px-GroupDiagramMiniD10.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/48px-GroupDiagramMiniD10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/GroupDiagramMiniD10.svg/64px-GroupDiagramMiniD10.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>11</th> <td>Z<sub>11</sub></td> <td>-</td> <td>可換、簡單</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC11.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/32px-GroupDiagramMiniC11.svg.png" decoding="async" width="32" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/48px-GroupDiagramMiniC11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/GroupDiagramMiniC11.svg/64px-GroupDiagramMiniC11.svg.png 2x" data-file-width="32" data-file-height="32" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="5">12</th> <td>Z<sub>12</sub> = Z<sub>4</sub> × Z<sub>3</sub></td> <td>Z<sub>6</sub> , Z<sub>4</sub> , Z<sub>3</sub> , Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/64px-GroupDiagramMiniC12.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/96px-GroupDiagramMiniC12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/GroupDiagramMiniC12.svg/128px-GroupDiagramMiniC12.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> Z<sub>2</sub> × Z<sub>6</sub> = Z<sub>2</sub> × Z<sub>2</sub> × Z<sub>3</sub> = Dih<sub>2</sub> × Z<sub>3</sub></td> <td>3 × Z<sub>6</sub>, Z<sub>3</sub>, Dih<sub>2</sub>, 3 × Z<sub>2</sub> </td><td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2C6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/GroupDiagramMiniC2C6.svg/64px-GroupDiagramMiniC2C6.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/GroupDiagramMiniC2C6.svg/96px-GroupDiagramMiniC2C6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/GroupDiagramMiniC2C6.svg/128px-GroupDiagramMiniC2C6.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> Dih<sub>6</sub> = Dih<sub>3</sub> × Z<sub>2</sub></td> <td>Z<sub>6</sub> , 2 × Dih<sub>3</sub> , 3 × Dih<sub>2</sub> , Z<sub>3</sub> , 7 × Z<sub>2</sub> </td><td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/64px-GroupDiagramMiniD12.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/96px-GroupDiagramMiniD12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMiniD12.svg/128px-GroupDiagramMiniD12.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> <i>A</i><sub>4</sub> </td> <td>Z<sub>2</sub><sup>2</sup>, 4 × Z<sub>3</sub>, 3 × Z<sub>2</sub> </td> <td>非可換；最小確定<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)" title="拉格朗日定理 (群論)">拉格朗日定理</a>之相反敘述不是對的群：沒有6階的子群</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniA4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/64px-GroupDiagramMiniA4.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/96px-GroupDiagramMiniA4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/GroupDiagramMiniA4.svg/128px-GroupDiagramMiniA4.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dic<sub>3</sub> = Z<sub>3</sub>和Z<sub>4</sub>的<a href="/wiki/%E5%8D%8A%E7%9B%B4%E7%A7%AF" title="半直积">半直積</a>，其中Z<sub>4</sub>以反演作用於Z<sub>3</sub>上</td> <td>Z<sub>2</sub>, Z<sub>3</sub>, 3 × Z<sub>4</sub>, Z<sub>6</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniX12.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/86px-GroupDiagramMiniX12.svg.png" decoding="async" width="86" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/129px-GroupDiagramMiniX12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniX12.svg/172px-GroupDiagramMiniX12.svg.png 2x" data-file-width="86" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>13</th> <td>Z<sub>13</sub> </td> <td>-</td> <td>可換、簡單</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC13.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/64px-GroupDiagramMiniC13.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/96px-GroupDiagramMiniC13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/GroupDiagramMiniC13.svg/128px-GroupDiagramMiniC13.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="2">14</th> <td>Z<sub>14</sub> = Z<sub>2</sub> × Z<sub>7</sub></td> <td> Z<sub>7</sub> , Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC14.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/64px-GroupDiagramMiniC14.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/96px-GroupDiagramMiniC14.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/GroupDiagramMiniC14.svg/128px-GroupDiagramMiniC14.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td> Dih<sub>7</sub></td> <td> Z<sub>7</sub> , 7 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD14.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/64px-GroupDiagramMiniD14.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/96px-GroupDiagramMiniD14.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/GroupDiagramMiniD14.svg/128px-GroupDiagramMiniD14.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th>15</th> <td>Z<sub>15</sub> = Z<sub>3</sub> × Z<sub>5</sub></td> <td>Z<sub>5</sub> , Z<sub>3</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC15.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/64px-GroupDiagramMiniC15.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/96px-GroupDiagramMiniC15.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC15.svg/128px-GroupDiagramMiniC15.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <th rowspan="14">16</th> <td>Z<sub>16</sub></td> <td>Z<sub>8</sub> , Z<sub>4</sub> , Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/64px-GroupDiagramMiniC16.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/96px-GroupDiagramMiniC16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/GroupDiagramMiniC16.svg/128px-GroupDiagramMiniC16.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>2</sub><sup>4</sup></td> <td>15 × Z<sub>2</sub>, 35 × Dih<sub>2</sub>, 15 × Z<sub>2</sub><sup>3</sup></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2x4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniC2x4.svg/64px-GroupDiagramMiniC2x4.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniC2x4.svg/96px-GroupDiagramMiniC2x4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/GroupDiagramMiniC2x4.svg/128px-GroupDiagramMiniC2x4.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>4</sub> × Z<sub>2</sub><sup>2</sup> </td><td>7 × Z<sub>2</sub>, 4 × Z<sub>4</sub>, 7 × Dih<sub>2</sub>, Z<sub>2</sub><sup>3</sup>, 6 × Z<sub>4</sub> × Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2x2C4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/78px-GroupDiagramMiniC2x2C4.svg.png" decoding="async" width="78" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/117px-GroupDiagramMiniC2x2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/156px-GroupDiagramMiniC2x2C4.svg.png 2x" data-file-width="78" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>8</sub> × Z<sub>2</sub></td> <td>3 × Z<sub>2</sub>, 2 × Z<sub>4</sub>, Dih<sub>2</sub>, 2 × Z<sub>8</sub>, Z<sub>4</sub> × Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2C8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/64px-GroupDiagramMiniC2C8.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/96px-GroupDiagramMiniC2C8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/128px-GroupDiagramMiniC2C8.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>4</sub><sup>2</sup></td> <td>3 × Z<sub>2</sub>, 6 × Z<sub>4</sub>, Dih<sub>2</sub>, 3 × Z<sub>4</sub> × Z<sub>2</sub></td> <td>可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC4x2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC4x2.svg/64px-GroupDiagramMiniC4x2.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC4x2.svg/96px-GroupDiagramMiniC4x2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC4x2.svg/128px-GroupDiagramMiniC4x2.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dih<sub>8</sub></td> <td>Z<sub>8</sub> , 2 × Dih<sub>4</sub> , 4 × Dih<sub>2</sub> , Z<sub>4</sub> , 9 × Z<sub>2</sub></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniD16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/64px-GroupDiagramMiniD16.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/96px-GroupDiagramMiniD16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/GroupDiagramMiniD16.svg/128px-GroupDiagramMiniD16.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Dih<sub>4</sub> × Z<sub>2</sub></td> <td> 2 × Dih<sub>4</sub> , Z<sub>4</sub> × Z<sub>2</sub> , 2 × Z<sub>2</sub><sup>3<sup>, 7 × Z<sub>2</sub><sup>2</sup> , 2 × Z<sub>4</sub> , 11 × Z<sub>2</sub> </sup></sup></td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2D8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/76px-GroupDiagramMiniC2D8.svg.png" decoding="async" width="76" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/114px-GroupDiagramMiniC2D8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/GroupDiagramMiniC2D8.svg/152px-GroupDiagramMiniC2D8.svg.png 2x" data-file-width="76" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><a href="/wiki/%E5%9B%9B%E5%85%83%E7%BE%A4" title="四元群">廣義四元群</a>, <i>Q</i><sub>16</sub> = Dic<sub>4</sub></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQ16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/119px-GroupDiagramMiniQ16.svg.png" decoding="async" width="119" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/179px-GroupDiagramMiniQ16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/GroupDiagramMiniQ16.svg/238px-GroupDiagramMiniQ16.svg.png 2x" data-file-width="119" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><i>Q</i><sub>8</sub> × Z<sub>2</sub></td> <td> </td> <td>非可換、<a href="/wiki/%E6%BC%A2%E5%BD%8C%E7%88%BE%E9%A0%93%E7%BE%A4" class="mw-redirect" title="漢彌爾頓群">漢彌爾頓群</a></td> <td><figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:GroupDiagramMiniC2Q8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/100px-GroupDiagramMiniC2Q8.svg.png" decoding="async" width="100" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/150px-GroupDiagramMiniC2Q8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/GroupDiagramMiniC2Q8.svg/200px-GroupDiagramMiniC2Q8.svg.png 2x" data-file-width="100" data-file-height="64" /></a><figcaption><i>wrong</i></figcaption></figure></td> </tr> <tr> <td>16階之<a href="/w/index.php?title=%E6%93%AC%E4%BA%8C%E9%9D%A2%E9%AB%94%E7%BE%A4&action=edit&redlink=1" class="new" title="擬二面體群（页面不存在）">擬二面體群</a></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniQH16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/84px-GroupDiagramMiniQH16.svg.png" decoding="async" width="84" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/126px-GroupDiagramMiniQH16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/GroupDiagramMiniQH16.svg/168px-GroupDiagramMiniQH16.svg.png 2x" data-file-width="84" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>16階之<a href="/w/index.php?title=%E6%A8%A1%E7%BE%A4&action=edit&redlink=1" class="new" title="模群（页面不存在）">模群</a></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2C8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/64px-GroupDiagramMiniC2C8.svg.png" decoding="async" width="64" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/96px-GroupDiagramMiniC2C8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/GroupDiagramMiniC2C8.svg/128px-GroupDiagramMiniC2C8.svg.png 2x" data-file-width="64" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>Z<sub>4</sub>和Z<sub>4</sub>的<a href="/w/index.php?title=%E5%8D%8A%E7%9B%B4%E5%92%8C&action=edit&redlink=1" class="new" title="半直和（页面不存在）">半直和</a>，其中一個以反演作用在另一個上</td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMinix3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/77px-GroupDiagramMinix3.svg.png" decoding="async" width="77" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/116px-GroupDiagramMinix3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/GroupDiagramMinix3.svg/154px-GroupDiagramMinix3.svg.png 2x" data-file-width="77" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td>由<a href="/wiki/%E6%B3%A1%E5%88%A9%E7%9F%A9%E9%99%A3" title="泡利矩陣">泡利矩陣</a>產生的群</td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniC2x2C4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/78px-GroupDiagramMiniC2x2C4.svg.png" decoding="async" width="78" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/117px-GroupDiagramMiniC2x2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GroupDiagramMiniC2x2C4.svg/156px-GroupDiagramMiniC2x2C4.svg.png 2x" data-file-width="78" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> <tr> <td><i>G</i><sub>4,4</sub></td> <td> </td> <td>非可換</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:GroupDiagramMiniG44.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/72px-GroupDiagramMiniG44.svg.png" decoding="async" width="72" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/108px-GroupDiagramMiniG44.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/GroupDiagramMiniG44.svg/144px-GroupDiagramMiniG44.svg.png 2x" data-file-width="72" data-file-height="64" /></a><figcaption></figcaption></figure></td> </tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="小群圖書館"><span id=".E5.B0.8F.E7.BE.A4.E5.9C.96.E6.9B.B8.E9.A4.A8"></span>小群圖書館</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=4" title="编辑章节：小群圖書館"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>群論<a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BB%A3%E6%95%B8%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="電腦代數系統">電腦代數系統</a><a href="/wiki/GAP" class="mw-redirect mw-disambig" title="GAP">GAP</a>包含著描述了「小」階之群的「小群圖書館」。這些群以<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%A7%8B" title="群同構">同構</a>為分列出。現在，這個圖書館已包含了下列個群： </p> <ul><li>至多2000階的群，除了1024階的(423 164 062個群)；</li> <li>5<sup>5</sup>階和7<sup>4</sup>階的群(92個群)；</li> <li><i>q</i><sup><i>n</i></sup>×<i>p</i>階的群，其中<i>q</i><sup><i>n</i></sup>整除2<sup>8</sup>、3<sup>6</sup>、5<sup>5</sup>或7<sup>4</sup>且<i>p</i>為不同於<i>q</i>的任意質數；</li> <li>因式分解成至多3個質數的群。</li></ul> <p>它包含著上述的群以電腦上可讀形式顯示之詳盡描述。 </p><p>這個圖書館由Hans Ulrich Besche、Bettina Eick和Eamonn O'Brien所建構及準備；見http://www.tu-bs.de/~hubesche/small.html。<sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<a href="/wiki/Wikipedia:%E5%A4%B1%E6%95%88%E9%93%BE%E6%8E%A5" title="Wikipedia:失效链接"><span title="自2017年12月失效">永久失效連結</span></a>]</span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="另見"><span id=".E5.8F.A6.E8.A6.8B"></span>另見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=5" title="编辑章节：另見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%E5%B0%8F%E6%8B%89%E4%B8%81%E6%96%B9%E9%99%A3%E5%92%8C%E6%93%AC%E7%BE%A4&action=edit&redlink=1" class="new" title="小拉丁方陣和擬群（页面不存在）">小拉丁方陣和擬群</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8F%E7%BE%A4%E5%88%97%E8%A1%A8&action=edit&section=6" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.math.usf.edu/~eclark/algctlg/small_groups.html">小群</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20061205230022/http://www.math.usf.edu/~eclark/algctlg/small_groups.html">页面存档备份</a>，存于<a 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