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<div class="ChapSects"><a href="chap3.html#X7EB517EB7A50A58F">3 <span class="Heading">The User Interface to the <strong class="pkg">GAP</strong> Character Table Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7A5EECF8853A05A0">3.1 <span class="Heading">Accessing Data of the <strong class="pkg">CTblLib</strong> Package</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X818A9DE5799A4809">3.1-1 <span class="Heading">Admissible Names for Character Tables in <strong class="pkg">CTblLib</strong>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86C06F408706F27A">3.1-2 CharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E42C22278DA4048">3.1-3 BrauerTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C091641852BB6FE">3.1-4 AllCharacterTableNames</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EC4A57E8393D75C">3.1-5 OneCharacterTableName</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X845DEEE282D4F1EB">3.1-6 NameOfEquivalentLibraryCharacterTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7EED71B4827BEBFA">3.2 <span class="Heading">The Interface to the <strong class="pkg">TomLib</strong> Package</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8400C0397CA2FA3C">3.2-1 TableOfMarks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87110C1584D09BE4">3.2-2 CharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A82CB487DBDDC53">3.2-3 FusionCharTableTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B1AAED68753B1BE">3.2-4 FusionToTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83D85EB2792D9D22">3.3 <span class="Heading">The Interface to <strong class="pkg">GAP</strong>'s Group Libraries</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78DCD38B7D96D8A4">3.3-1 GroupInfoForCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87CFDC8081767CA4">3.3-2 KnowsSomeGroupInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78B7A5DB87B5285C">3.3-3 CharacterTableForGroupInfo</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83B9EFB47CE0A617">3.3-5 GroupForTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8032EF83823B1D88">3.3-6 AtlasStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7AADD757809DFDD5">3.3-7 IsNontrivialDirectProduct</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C1F6C01852D7933">3.4 <span class="Heading">Unipotent Characters of Finite Groups of Lie Type</span></a>
</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79F645F278C27F23">3.4-1 UnipotentCharacter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X803FD7F87AFBBDE4">3.4-2 DeligneLusztigNames</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84B9D8EF84EB4145">3.4-3 DeligneLusztigName</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7900893987A89915">3.5 <span class="Heading"><strong class="pkg">Browse</strong> Applications Provided by
<strong class="pkg">CTblLib</strong></span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8684C73E844E7033">3.5-1 DisplayCTblLibInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A038A267CD17032">3.5-2 BrowseCTblLibInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83FDD44E7E0AE885">3.5-3 BrowseCommonIrrationalities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87CA74CF7B533CC6">3.5-4 BrowseCTblLibDifferences</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8135745381E1C0E2">3.5-5 BrowseAtlasContents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X810789F77F20C839">3.5-6 DisplayAtlasContents</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79E3F783818FE75A">3.5-10 BrowseAtlasImprovements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E5CFB187E313B11">3.6 <span class="Heading">Duplicates of Library Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FD4A9DF7CBAA418">3.6-1 IsDuplicateTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X860F49407882658F">3.6-2 IdentifierOfMainTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C83435B78DE62F6">3.6-3 IdentifiersOfDuplicateTables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7CCFEC998135F6FA">3.7 <span class="Heading">Attributes for Library Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8150E63F7DBDF252">3.7-1 Maxes</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82DC2E7779322DA8">3.7-2 ProjectivesInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82A008987DB887C2">3.7-3 ExtensionInfoCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X851118377D1D6EC9">3.7-4 ConstructionInfoCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X871562FD7F982C12">3.7-5 InfoText</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X867813EB79DC6953">3.7-6 IsAtlasCharacterTable</a></span>
</div></div>
</div>
<h3>3 <span class="Heading">The User Interface to the <strong class="pkg">GAP</strong> Character Table Library</span></h3>
<p><a id="X7A5EECF8853A05A0" name="X7A5EECF8853A05A0"></a></p>
<h4>3.1 <span class="Heading">Accessing Data of the <strong class="pkg">CTblLib</strong> Package</span></h4>
<p><a id="X818A9DE5799A4809" name="X818A9DE5799A4809"></a></p>
<h5>3.1-1 <span class="Heading">Admissible Names for Character Tables in <strong class="pkg">CTblLib</strong>
</span></h5>
<p>When you access a character table from the <strong class="pkg">GAP</strong> Character Table Library, this table is specified by an admissible name.</p>
<p>Admissible names for the <em>ordinary character table</em> <span class="SimpleMath">tbl</span> of the group <span class="SimpleMath">G</span> are</p>
<ul>
<li><p>an <strong class="pkg">Atlas</strong> like name if <span class="SimpleMath">tbl</span> is an <strong class="pkg">Atlas</strong> table (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>), for example <code class="code">"M22"</code> for the table of the Mathieu group <span class="SimpleMath">M_22</span>, <code class="code">"L2(13).2"</code> for <span class="SimpleMath">L_2(13):2</span>, and <code class="code">"12_1.U4(3).2_1"</code> for <span class="SimpleMath">12_1.U_4(3).2_1</span>.</p>
<p>(The difference to the name printed in the <strong class="pkg">Atlas</strong> is that subscripts and superscripts are omitted except if they are used to qualify integer values, and double dots are replaced by a single dot.)</p>
</li>
<li><p>the names that were admissible for tables of <span class="SimpleMath">G</span> in the <strong class="pkg">CAS</strong> system if the <strong class="pkg">CAS</strong> table library contained a table of <span class="SimpleMath">G</span>, for example <code class="code">sl42</code> for the table of the alternating group <span class="SimpleMath">A_8</span>.</p>
<p>(But note that the ordering of rows and columns of the <strong class="pkg">GAP</strong> table may be different from that in <strong class="pkg">CAS</strong>, see Section <a href="chap4.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a>.)</p>
</li>
<li><p>some <q>relative</q> names, as follows.</p>
<ul>
<li><p>If <span class="SimpleMath">G</span> is the <span class="SimpleMath">n</span>-th maximal subgroup (in decreasing group order) of a group whose library table <span class="SimpleMath">subtbl</span> is available in <strong class="pkg">GAP</strong> and stores the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) value, and if <code class="code">name</code> is an admissible name for <span class="SimpleMath">subtbl</span> then <code class="code">name</code>M<span class="SimpleMath">n</span> is admissible for <span class="SimpleMath">tbl</span>. For example, the name <code class="code">"J3M2"</code> can be used to access the second maximal subgroup of the sporadic simple Janko group <span class="SimpleMath">J_3</span> which has the admissible name <code class="code">"J3"</code>.</p>
</li>
<li><p>If <span class="SimpleMath">G</span> is a nontrivial Sylow <span class="SimpleMath">p</span> normalizer in a sporadic simple group with admissible name <code class="code">name</code> –where nontrivial means that <span class="SimpleMath">G</span> is not isomorphic to a subgroup of <span class="SimpleMath">p:(p-1)</span>– then <code class="code">name</code>N<span class="SimpleMath">p</span> is an admissible name of <span class="SimpleMath">tbl</span>. For example, the name <code class="code">"J4N11"</code> can be used to access the table of the Sylow <span class="SimpleMath">11</span> normalizer in the sporadic simple Janko group <span class="SimpleMath">J_4</span>.</p>
</li>
<li><p>In a few cases, the table of the Sylow <span class="SimpleMath">p</span>-subgroup of <span class="SimpleMath">G</span> is accessible via the name <code class="code">name</code>Syl<span class="SimpleMath">p</span> where <code class="code">name</code> is an admissible name of the table of <span class="SimpleMath">G</span>. For example, <code class="code">"A11Syl2"</code> is an admissible name for the table of the Sylow <span class="SimpleMath">2</span>-subgroup of the alternating group <span class="SimpleMath">A_11</span>.</p>
</li>
<li><p>In a few cases, the table of an element centralizer in <span class="SimpleMath">G</span> is accessible via the name <code class="code">name</code>C<span class="SimpleMath">cl</span> where <code class="code">name</code> is an admissible name of the table of <span class="SimpleMath">G</span>. For example, <code class="code">"M11C2"</code> is an admissible name for the table of an involution centralizer in the Mathieu group <span class="SimpleMath">M_11</span>.</p>
</li>
</ul>
</li>
</ul>
<p>The recommended way to access a <em>Brauer table</em> is via applying the <code class="keyw">mod</code> operator to the ordinary table and the desired characteristic (see <code class="func">BrauerTable</code> (<a href="../../../doc/ref/chap71.html#X8476B25A79D7A7FC"><span class="RefLink">Reference: BrauerTable</span></a>) and Section <a href="../../../doc/ref/chap71.html#X7CADCBC9824CB624"><span class="RefLink">Reference: Operators for Character Tables</span></a>), so it is not necessary to define admissible names of Brauer tables.</p>
<p>A <em>generic character table</em> (see Section <a href="chap4.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a>) is accessible only by the name given by its <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value.</p>
<p><a id="X86C06F408706F27A" name="X86C06F408706F27A"></a></p>
<h5>3.1-2 CharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">tblname</var>[, <var class="Arg">para1</var>[, <var class="Arg">para2</var>]] )</td><td class="tdright">( method )</td></tr></table></div>
<p>If the only argument is a string <var class="Arg">tblname</var> and if this is an admissible name (see <a href="chap3.html#X818A9DE5799A4809"><span class="RefLink">3.1-1</span></a>) of a library character table then <code class="func">CharacterTable</code> returns this library table, otherwise <code class="keyw">fail</code>.</p>
<p>If <code class="func">CharacterTable</code> is called with more than one argument then the first must be a string <var class="Arg">tblname</var> specifying a series of groups which is implemented via a generic character table, for example <code class="code">"Symmetric"</code> for symmetric groups; the remaining arguments specialize the desired member of the series (see Section <a href="chap4.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a> for a list of available generic tables). If no generic table with name <var class="Arg">tblname</var> is available or if the parameters are not admissible then <code class="func">CharacterTable</code> returns <code class="keyw">fail</code>.</p>
<p>A call of <code class="func">CharacterTable</code> may cause that some library files are read and that some table objects are constructed from the data stored in these files, so fetching a library table may take more time than one expects.</p>
<p>Case is not significant for <var class="Arg">tblname</var>. For example, both <code class="code">"suzm3"</code> and <code class="code">"SuzM3"</code> can be entered in order to access the character table of the third class of maximal subgroups of the sporadic simple Suzuki group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5:= CharacterTable( "A5.2" );</span>
CharacterTable( "A5.2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">sym5:= CharacterTable( "Symmetric", 5 );</span>
CharacterTable( "Sym(5)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( s5, sym5 );</span>
rec( columns := (2,3,4,7,5), group := Group(()),
rows := (1,7,3,4,6,5,2) )
</pre></div>
<p>The above two tables are tables of the symmetric group on five letters; the first is in <strong class="pkg">Atlas</strong> format (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>), the second is constructed from the generic table for symmetric groups (see <a href="chap4.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "J5" );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "A5" ) mod 2;</span>
BrauerTable( "A5", 2 )
</pre></div>
<p><a id="X7E42C22278DA4048" name="X7E42C22278DA4048"></a></p>
<h5>3.1-3 BrauerTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrauerTable</code>( <var class="Arg">tblname</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Called with a string <var class="Arg">tblname</var> and a prime integer <var class="Arg">p</var>, <code class="func">BrauerTable</code> returns the <var class="Arg">p</var>-modular character table of the ordinary character table with admissible name <var class="Arg">tblname</var>, if such an ordinary character table exists and if <strong class="pkg">GAP</strong> can compute its <var class="Arg">p</var>-modular table. Otherwise <code class="keyw">fail</code> is returned.</p>
<p>The default method delegates to <code class="func">BrauerTable</code> (<a href="../../../doc/ref/chap71.html#X8476B25A79D7A7FC"><span class="RefLink">Reference: BrauerTable for a character table, and a prime integer</span></a>) with arguments the <code class="func">CharacterTable</code> (<a href="chap3.html#X86C06F408706F27A"><span class="RefLink">3.1-2</span></a>) value of <var class="Arg">tblname</var> and <var class="Arg">p</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( "A5", 2 );</span>
BrauerTable( "A5", 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( "J5", 2 ); # no ordinary table with name J5</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( "M", 2 ); # Brauer table not known</span>
fail
</pre></div>
<p><a id="X7C091641852BB6FE" name="X7C091641852BB6FE"></a></p>
<h5>3.1-4 AllCharacterTableNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllCharacterTableNames</code>( [<var class="Arg">func</var>, <var class="Arg">val</var>, <var class="Arg">...</var>[, <var class="Arg">OfThose</var>, <var class="Arg">func</var>]]<var class="Arg">:</var> <var class="Arg">OrderedBy</var> <var class="Arg">:=</var> <var class="Arg">func</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Similar to group libraries (see Chapter <a href="../../../doc/ref/chap50.html#X81B00B667D2BD022"><span class="RefLink">Reference: Group Libraries</span></a>), the <strong class="pkg">GAP</strong> Character Table Library can be used to search for ordinary character tables with prescribed properties.</p>
<p>A specific library table can be selected by an admissible name, see <a href="chap3.html#X818A9DE5799A4809"><span class="RefLink">3.1-1</span></a>.</p>
<p>The <em>selection function</em> (see <a href="../../../doc/ref/chap50.html#X82676ED5826E9E2E"><span class="RefLink">Reference: Selection Functions</span></a>) for character tables from the <strong class="pkg">GAP</strong> Character Table Library that have certain abstract properties is <code class="func">AllCharacterTableNames</code>. Contrary to the situation in the case of group libraries, the selection function returns a list not of library character tables but of their names; using <code class="func">CharacterTable</code> (<a href="chap3.html#X86C06F408706F27A"><span class="RefLink">3.1-2</span></a>) one can then access the tables themselves.</p>
<p><code class="func">AllCharacterTableNames</code> takes an arbitrary even number of arguments. The argument at each odd position must be a function, and the argument at the subsequent even position must be either a value that this function must return when called for the character table in question, in order to have the name of the table included in the selection, or a list of such values, or a function that returns <code class="keyw">true</code> for such a value, and <code class="keyw">false</code> otherwise. For example,</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">names:= AllCharacterTableNames();;</span>
</pre></div>
<p>returns a list containing one admissible name of each ordinary character table in the <strong class="pkg">GAP</strong> library,</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">simpnames:= AllCharacterTableNames( IsSimple, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsAbelian, false );;</span>
</pre></div>
<p>returns a list containing an admissible name of each ordinary character table in the <strong class="pkg">GAP</strong> library whose groups are nonabelian and simple, and</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( IsSimple, true, IsAbelian, false,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [ 1 .. 100 ] );</span>
[ "A5", "A6M2", "Alt(5)" ]
</pre></div>
<p>returns a list containing an admissible name of each ordinary character table in the <strong class="pkg">GAP</strong> library whose groups are nonabelian and simple and have order at most <span class="SimpleMath">100</span>, respectively. (Note that <code class="code">"A5"</code>, <code class="code">"A6M2"</code>, and <code class="code">"Alt(5)"</code> are identifiers of permutation equivalent character tables. It would be possible to exclude duplicates, see Section <a href="chap3.html#X7E5CFB187E313B11"><span class="RefLink">3.6</span></a>).</p>
<p>Similarly,</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( Size, IsPrimeInt );</span>
[ "2.Alt(2)", "Alt(3)", "C2", "C3", "Sym(2)" ]
</pre></div>
<p>returns the list of all identifiers of library tables whose <code class="func">Size</code> (<a href="../../../doc/ref/chap30.html#X858ADA3B7A684421"><span class="RefLink">Reference: Size</span></a>) value is a prime integer, and</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( Identifier,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> PositionSublist( x, "L8" ) <> fail );</span>
[ "L8(2)", "P1L82", "P2L82" ]
</pre></div>
<p>returns the identifiers that contain the string <code class="code">"L8"</code> as a substring.</p>
<p>For the sake of efficiency, the attributes whose names are listed in <code class="code">CTblLib.SupportedAttributes</code> are handled in a special way, <strong class="pkg">GAP</strong> need not read all files of the table library in these cases in order to find the desired names.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CTblLib.SupportedAttributes;</span>
[ "AbelianInvariants", "HasFusionToTom", "Identifier",
"IdentifiersOfDuplicateTables", "InfoText", "IsAbelian",
"IsAlmostSimple", "IsAtlasCharacterTable", "IsDuplicateTable",
"IsNontrivialDirectProduct", "IsPerfect", "IsQuasisimple",
"IsSimple", "IsSporadicSimple", "KnowsDeligneLusztigNames",
"KnowsSomeGroupInfo", "Maxes", "NamesOfFusionSources",
"NrConjugacyClasses", "Size" ]
</pre></div>
<p>If the <strong class="pkg">Browse</strong> package (see <a href="chapBib.html#biBBrowse">[BL23]</a>) is not loaded then <code class="code">CTblLib.SupportedAttributes</code> contains only <code class="code">"Identifier"</code>, and <code class="func">AllCharacterTableNames</code> will be very slow when one selects character tables according to other attributes from the list shown above.</p>
<p>The global option <code class="code">OrderedBy</code> can be used to prescribe the ordering of the result. The value of this option, if given, must be a function that takes a character table as its unique argument; the result list is then sorted according to the results of this function (w. r. t. the comparison by <strong class="pkg">GAP</strong>'s <code class="code">\<</code> operation).</p>
<p>For example, we may be interested in the tables of small sporadic simple groups, ordered alphabetically or by size (<code class="func">Size</code> (<a href="../../../doc/ref/chap71.html#X81EFD9FE804AC6EE"><span class="RefLink">Reference: Size for a character table</span></a>)) or by the number of conjugacy classes (<code class="func">NrConjugacyClasses</code> (<a href="../../../doc/ref/chap71.html#X81EFD9FE804AC6EE"><span class="RefLink">Reference: NrConjugacyClasses for a character table</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( IsSporadicSimple, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [ 1 .. 10^6 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsDuplicateTable, false );</span>
[ "J1", "J2", "M11", "M12", "M22" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( IsSporadicSimple, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [ 1 .. 10^6 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsDuplicateTable, false : OrderedBy:= Size );</span>
[ "M11", "M12", "J1", "M22", "J2" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( IsSporadicSimple, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [ 1 .. 10^6 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsDuplicateTable, false : OrderedBy:= NrConjugacyClasses );</span>
[ "M11", "M22", "J1", "M12", "J2" ]
</pre></div>
<p>(Note that the alphabtical ordering could also be achieved by entering <code class="code">OrderedBy:= Identifier</code>.)</p>
<p>If the dummy function <code class="code">OfThose</code> is an argument at an odd position then the following argument <var class="Arg">func</var> must be a function that takes a character table and returns a name of a character table or a list of names; this is interpreted as replacement of the names computed up to this position by the union of names returned by <var class="Arg">func</var>. For example, <var class="Arg">func</var> may be <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) or <code class="func">NamesOfFusionSources</code> (<a href="../../../doc/ref/chap73.html#X7F6569D5786A9D49"><span class="RefLink">Reference: NamesOfFusionSources</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxesnames:= AllCharacterTableNames( IsSporadicSimple, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> HasMaxes, true,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> OfThose, Maxes );;</span>
</pre></div>
<p>returns the union of names of ordinary tables of those maximal subgroups of sporadic simple groups that are contained in the table library in the sense that the attribute <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) is set.</p>
<p>For the sake of efficiency, <code class="code">OfThose</code> followed by one of the arguments <code class="func">AutomorphismGroup</code> (<a href="../../../doc/ref/chap40.html#X87677B0787B4461A"><span class="RefLink">Reference: AutomorphismGroup</span></a>), <code class="func">SchurCover</code> (<a href="../../../doc/ref/chap39.html#X7DD1E37987612042"><span class="RefLink">Reference: SchurCover</span></a>), <code class="code">CompleteGroup</code> is handled in a special way.</p>
<p><a id="X7EC4A57E8393D75C" name="X7EC4A57E8393D75C"></a></p>
<h5>3.1-5 OneCharacterTableName</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCharacterTableName</code>( [<var class="Arg">func</var>, <var class="Arg">val</var>, <var class="Arg">...</var>[, <var class="Arg">OfThose</var>, <var class="Arg">func</var>]]<var class="Arg">:</var> <var class="Arg">OrderedBy</var> <var class="Arg">:=</var> <var class="Arg">func</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The example function for character tables from the <strong class="pkg">GAP</strong> Character Table Library that have certain abstract properties is <code class="func">OneCharacterTableName</code>. It is analogous to the selection function <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>), the difference is that it returns one <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of a character table with the properties in question instead of the list of all such values. If no table with the required properties is contained in the <strong class="pkg">GAP</strong> Character Table Library then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneCharacterTableName( IsSimple, true, Size, 60 );</span>
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">OneCharacterTableName( IsSimple, true, Size, 20 );</span>
fail
</pre></div>
<p>The global option <code class="code">OrderedBy</code> can be used to search for a <q>smallest</q> example, according to the value of the option. If this function is one of the attributes whose names are listed in <code class="code">CTblLib.SupportedAttributes</code> then the tables are processed according to increasing values of the option, which may speed up the search.</p>
<p><a id="X845DEEE282D4F1EB" name="X845DEEE282D4F1EB"></a></p>
<h5>3.1-6 NameOfEquivalentLibraryCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NameOfEquivalentLibraryCharacterTable</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NamesOfEquivalentLibraryCharacterTables</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">ordtbl</var> be an ordinary character table. <code class="func">NameOfEquivalentLibraryCharacterTable</code> returns the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of a character table in the <strong class="pkg">GAP</strong> Character Table Library that is permutation equivalent to <var class="Arg">ordtbl</var> (see <code class="func">TransformingPermutationsCharacterTables</code> (<a href="../../../doc/ref/chap71.html#X849731AA7EC9FA73"><span class="RefLink">Reference: TransformingPermutationsCharacterTables</span></a>)) if such a character table exists, and <code class="keyw">fail</code> otherwise. <code class="func">NamesOfEquivalentLibraryCharacterTables</code> returns the list of all <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) values of character tables in the <strong class="pkg">GAP</strong> Character Table Library that are permutation equivalent to <var class="Arg">ordtbl</var>; thus an empty list is returned in this case if no equivalent library table exists.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Alternating", 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NameOfEquivalentLibraryCharacterTable( tbl );</span>
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">NamesOfEquivalentLibraryCharacterTables( tbl );</span>
[ "A5", "A6M2", "Alt(5)" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Cyclic", 17 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NameOfEquivalentLibraryCharacterTable( tbl );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">NamesOfEquivalentLibraryCharacterTables( tbl );</span>
[ ]
</pre></div>
<p><a id="X7EED71B4827BEBFA" name="X7EED71B4827BEBFA"></a></p>
<h4>3.2 <span class="Heading">The Interface to the <strong class="pkg">TomLib</strong> Package</span></h4>
<p>The <strong class="pkg">GAP</strong> Character Table Library contains ordinary character tables of all groups for which the <strong class="pkg">TomLib</strong> package <a href="chapBib.html#biBTomLib">[MNP19]</a> contains the table of marks. This section describes the mapping between these character tables and their tables of marks.</p>
<p>If the <strong class="pkg">TomLib</strong> package is not loaded then <code class="func">FusionToTom</code> (<a href="chap3.html#X7B1AAED68753B1BE"><span class="RefLink">3.2-4</span></a>) is the only available function from this section, but of course it is of little interest in this situation.</p>
<p><a id="X8400C0397CA2FA3C" name="X8400C0397CA2FA3C"></a></p>
<h5>3.2-1 TableOfMarks</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarks</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be an ordinary character table from the <strong class="pkg">GAP</strong> Character Table Library, for the group <span class="SimpleMath">G</span>, say. If the <strong class="pkg">TomLib</strong> package is loaded and contains the table of marks of <span class="SimpleMath">G</span> then there is a method based on <code class="func">TableOfMarks</code> (<a href="../../../doc/ref/chap70.html#X85B262AB7E219C34"><span class="RefLink">Reference: TableOfMarks for a string</span></a>) that returns this table of marks. If there is no such table of marks but <var class="Arg">tbl</var> knows its underlying group then this method delegates to the group. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TableOfMarks( CharacterTable( "A5" ) );</span>
TableOfMarks( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">TableOfMarks( CharacterTable( "M" ) );</span>
fail
</pre></div>
<p><a id="X87110C1584D09BE4" name="X87110C1584D09BE4"></a></p>
<h5>3.2-2 CharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">tom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, this method for <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable for a group</span></a>) returns the character table corresponding to <var class="Arg">tom</var>.</p>
<p>If <var class="Arg">tom</var> comes from the <strong class="pkg">TomLib</strong> package, the character table comes from the <strong class="pkg">GAP</strong> Character Table Library. Otherwise, if <var class="Arg">tom</var> stores an <code class="func">UnderlyingGroup</code> (<a href="../../../doc/ref/chap70.html#X81E41D3880FA6C4C"><span class="RefLink">Reference: UnderlyingGroup for tables of marks</span></a>) value then the task is delegated to a <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable for a group</span></a>) method for this group, and if no underlying group is available then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( TableOfMarks( "A5" ) );</span>
CharacterTable( "A5" )
</pre></div>
<p><a id="X7A82CB487DBDDC53" name="X7A82CB487DBDDC53"></a></p>
<h5>3.2-3 FusionCharTableTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionCharTableTom</code>( <var class="Arg">tbl</var>, <var class="Arg">tom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be an ordinary character table from the <strong class="pkg">GAP</strong> Character Table Library with the attribute <code class="func">FusionToTom</code> (<a href="chap3.html#X7B1AAED68753B1BE"><span class="RefLink">3.2-4</span></a>), and let <var class="Arg">tom</var> be the table of marks from the <strong class="pkg">GAP</strong> package <strong class="pkg">TomLib</strong> that corresponds to <var class="Arg">tbl</var>. In this case, a method for <code class="func">FusionCharTableTom</code> (<a href="../../../doc/ref/chap70.html#X7A82CB487DBDDC53"><span class="RefLink">Reference: FusionCharTableTom</span></a>) is available that returns the fusion from <var class="Arg">tbl</var> to <var class="Arg">tom</var> that is given by the <code class="func">FusionToTom</code> (<a href="chap3.html#X7B1AAED68753B1BE"><span class="RefLink">3.2-4</span></a>) value of <var class="Arg">tbl</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );</span>
CharacterTable( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( "A5" );</span>
TableOfMarks( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">FusionCharTableTom( tbl, tom );</span>
[ 1, 2, 3, 5, 5 ]
</pre></div>
<p><a id="X7B1AAED68753B1BE" name="X7B1AAED68753B1BE"></a></p>
<h5>3.2-4 FusionToTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionToTom</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If this attribute is set for an ordinary character table <var class="Arg">tbl</var> then the <strong class="pkg">GAP</strong> Library of Tables of Marks contains the table of marks of the group of <var class="Arg">tbl</var>, and the attribute value is a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap70.html#X810E53597B5BB4F8"><span class="RefLink">Reference: Identifier for tables of marks</span></a>) component of the table of marks of <var class="Arg">tbl</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">map</code></strong></dt>
<dd><p>the fusion map,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code> (optional)</strong></dt>
<dd><p>a string describing the status of the fusion, and</p>
</dd>
<dt><strong class="Mark"><code class="code">perm</code> (optional)</strong></dt>
<dd><p>a permutation that establishes the bijection between the classes of maximal subgroups in the table of marks (see <code class="func">MaximalSubgroupsTom</code> (<a href="../../../doc/ref/chap70.html#X8325811586C00ECF"><span class="RefLink">Reference: MaximalSubgroupsTom</span></a>)) and the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) list of <var class="Arg">tbl</var>. Applying the permutation to the sublist of permutation characters (see <code class="func">PermCharsTom</code> (<a href="../../../doc/ref/chap70.html#X8016499282F0BA37"><span class="RefLink">Reference: PermCharsTom via fusion map</span></a>)) at the positions of the maximal subgroups of the table of marks yields the list of primitive permutation characters computed from the character tables described by the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) list. Usually, there is no <code class="code">perm</code> component, which means that the two lists of primitive permutation characters are equal. See Section <a href="chap2.html#X83BF83D87BC1B123"><span class="RefLink">2.3-5</span></a> for an example.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FusionToTom( CharacterTable( "2.A6" ) );</span>
rec( map := [ 1, 2, 5, 4, 8, 3, 7, 11, 11, 6, 13, 6, 13 ],
name := "2.A6", perm := (4,5),
text := "fusion map is unique up to table autom." )
</pre></div>
<p><a id="X8435BFE878FFFDAB" name="X8435BFE878FFFDAB"></a></p>
<h5>3.2-5 NameOfLibraryCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NameOfLibraryCharacterTable</code>( <var class="Arg">tomname</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the character table corresponding to the table of marks with <code class="func">Identifier</code> (<a href="../../../doc/ref/chap70.html#X810E53597B5BB4F8"><span class="RefLink">Reference: Identifier for tables of marks</span></a>) value <var class="Arg">tomname</var>. If no such character table exists in the <strong class="pkg">GAP</strong> Character Table Library or if the <strong class="pkg">TomLib</strong> package is not loaded then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NameOfLibraryCharacterTable( "A5" );</span>
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">NameOfLibraryCharacterTable( "S5" );</span>
"A5.2"
</pre></div>
<p><a id="X83D85EB2792D9D22" name="X83D85EB2792D9D22"></a></p>
<h4>3.3 <span class="Heading">The Interface to <strong class="pkg">GAP</strong>'s Group Libraries</span></h4>
<p>Sometimes it is useful to extend a character-theoretic computation with computations involving a group that has the character table in question. For many character tables in the <strong class="pkg">GAP</strong> Character Table Library, corresponding groups can be found in the various group libraries that are distributed with <strong class="pkg">GAP</strong>. This section describes how one can access the library groups that belong to a given character table.</p>
<p><a id="X78DCD38B7D96D8A4" name="X78DCD38B7D96D8A4"></a></p>
<h5>3.3-1 GroupInfoForCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupInfoForCharacterTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be an ordinary character table from the <strong class="pkg">GAP</strong> Character Table Library. <code class="func">GroupInfoForCharacterTable</code> returns a sorted list of pairs such that calling <code class="func">GroupForGroupInfo</code> (<a href="chap3.html#X804D2AC37E0D4755"><span class="RefLink">3.3-4</span></a>) with any of these pairs yields a group whose ordinary character table is <var class="Arg">tbl</var>, up to permutations of rows and columns.</p>
<p>Note that this group is in general <em>not</em> determined up to isomorphism, since nonisomorphic groups may have the same character table (including power maps).</p>
<p>Contrary to the attribute <code class="func">UnderlyingGroup</code> (<a href="../../../doc/ref/chap70.html#X81E41D3880FA6C4C"><span class="RefLink">Reference: UnderlyingGroup for tables of marks</span></a>), the entries of the <code class="func">GroupInfoForCharacterTable</code> list for <var class="Arg">tbl</var> are not related to the ordering of the conjugacy classes in <var class="Arg">tbl</var>.</p>
<p>Sources for this attribute are the <strong class="pkg">GAP</strong> databases of groups described in Chapter <a href="../../../doc/ref/chap50.html#X81B00B667D2BD022"><span class="RefLink">Reference: Group Libraries</span></a>, and the packages <strong class="pkg">AtlasRep</strong> and <strong class="pkg">TomLib</strong>, see also <code class="func">GroupForTom</code> (<a href="chap3.html#X83B9EFB47CE0A617"><span class="RefLink">3.3-5</span></a>) and <code class="func">AtlasStabilizer</code> (<a href="chap3.html#X8032EF83823B1D88"><span class="RefLink">3.3-6</span></a>). If these packages are not loaded then part of the information may be missing. If the <strong class="pkg">Browse</strong> package (see <a href="chapBib.html#biBBrowse">[BL23]</a>) is not loaded then <code class="func">GroupInfoForCharacterTable</code> returns always an empty list.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupInfoForCharacterTable( CharacterTable( "A5" ) );</span>
[ [ "AlternatingGroup", [ 5 ] ], [ "AtlasGroup", [ "A5" ] ],
[ "AtlasStabilizer", [ "A6", "A6G1-p6aB0" ] ],
[ "AtlasStabilizer", [ "A6", "A6G1-p6bB0" ] ],
[ "AtlasStabilizer", [ "L2(11)", "L211G1-p11aB0" ] ],
[ "AtlasStabilizer", [ "L2(11)", "L211G1-p11bB0" ] ],
[ "AtlasStabilizer", [ "L2(19)", "L219G1-p57aB0" ] ],
[ "AtlasStabilizer", [ "L2(19)", "L219G1-p57bB0" ] ],
[ "AtlasSubgroup", [ "A5.2", 1 ] ], [ "AtlasSubgroup", [ "A6", 1 ] ]
, [ "AtlasSubgroup", [ "A6", 2 ] ],
[ "AtlasSubgroup", [ "J2", 9 ] ],
[ "AtlasSubgroup", [ "L2(109)", 4 ] ],
[ "AtlasSubgroup", [ "L2(109)", 5 ] ],
[ "AtlasSubgroup", [ "L2(11)", 1 ] ],
[ "AtlasSubgroup", [ "L2(11)", 2 ] ],
[ "AtlasSubgroup", [ "S6(3)", 11 ] ],
[ "GroupForTom", [ "2^4:A5", 68 ] ],
[ "GroupForTom", [ "2^4:A5`", 56 ] ], [ "GroupForTom", [ "A5" ] ],
[ "GroupForTom", [ "A5xA5", 85 ] ], [ "GroupForTom", [ "A6", 21 ] ],
[ "GroupForTom", [ "J2", 99 ] ],
[ "GroupForTom", [ "L2(109)", 25 ] ],
[ "GroupForTom", [ "L2(11)", 15 ] ],
[ "GroupForTom", [ "L2(125)", 18 ] ],
[ "GroupForTom", [ "L2(16)", 18 ] ],
[ "GroupForTom", [ "L2(19)", 17 ] ],
[ "GroupForTom", [ "L2(29)", 19 ] ],
[ "GroupForTom", [ "L2(31)", 25 ] ],
[ "GroupForTom", [ "S5", 18 ] ], [ "PSL", [ 2, 4 ] ],
[ "PSL", [ 2, 5 ] ], [ "PerfectGroup", [ 60, 1 ] ],
[ "PrimitiveGroup", [ 5, 4 ] ], [ "PrimitiveGroup", [ 6, 1 ] ],
[ "PrimitiveGroup", [ 10, 1 ] ], [ "SmallGroup", [ 60, 5 ] ],
[ "TransitiveGroup", [ 5, 4 ] ], [ "TransitiveGroup", [ 6, 12 ] ],
[ "TransitiveGroup", [ 10, 7 ] ], [ "TransitiveGroup", [ 12, 33 ] ],
[ "TransitiveGroup", [ 15, 5 ] ], [ "TransitiveGroup", [ 20, 15 ] ],
[ "TransitiveGroup", [ 30, 9 ] ] ]
</pre></div>
<p><a id="X87CFDC8081767CA4" name="X87CFDC8081767CA4"></a></p>
<h5>3.3-2 KnowsSomeGroupInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KnowsSomeGroupInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var>, this function returns <code class="keyw">true</code> if the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>) is nonempty, and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">KnowsSomeGroupInfo( CharacterTable( "A5" ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">KnowsSomeGroupInfo( CharacterTable( "M" ) );</span>
false
</pre></div>
<p><a id="X78B7A5DB87B5285C" name="X78B7A5DB87B5285C"></a></p>
<h5>3.3-3 CharacterTableForGroupInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableForGroupInfo</code>( <var class="Arg">info</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function is a partial inverse of <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>). If <var class="Arg">info</var> has the form <code class="code">[ </code><span class="SimpleMath">funcname</span><code class="code">, </code><span class="SimpleMath">args</span><code class="code"> ]</code> and occurs in the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>) when called with a character table <span class="SimpleMath">t</span>, say, then <code class="func">CharacterTableForGroupInfo</code> returns a character table from the <strong class="pkg">GAP</strong> Character Table that is equivalent to <span class="SimpleMath">t</span>. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTableForGroupInfo( [ "AlternatingGroup", [ 5 ] ] );</span>
CharacterTable( "A5" )
</pre></div>
<p><a id="X804D2AC37E0D4755" name="X804D2AC37E0D4755"></a></p>
<h5>3.3-4 GroupForGroupInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupForGroupInfo</code>( <var class="Arg">info</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">info</var> has the form <code class="code">[ </code><span class="SimpleMath">funcname</span><code class="code">, </code><span class="SimpleMath">args</span><code class="code"> ]</code> and occurs in the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>) when called with a character table <span class="SimpleMath">tbl</span>, say, then <code class="func">GroupForGroupInfo</code> returns a group that is described by <var class="Arg">info</var> and whose character table is equal to <span class="SimpleMath">tbl</span>, up to permutations of rows and columns. Otherwise <code class="keyw">fail</code> is returned.</p>
<p>Typically, <span class="SimpleMath">funcname</span> is a string that is the name of a global <strong class="pkg">GAP</strong> function <span class="SimpleMath">fun</span>, say, and <span class="SimpleMath">args</span> is a list of arguments for this function such that <code class="code">CallFuncList( </code><span class="SimpleMath">fun</span><code class="code">, </code><span class="SimpleMath">args</span><code class="code"> )</code> yields the desired group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupForGroupInfo( [ "AlternatingGroup", [ 5 ] ] );</span>
Alt( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupForGroupInfo( [ "PrimitiveGroup", [ 5, 4 ] ] );</span>
A(5)
</pre></div>
<p><a id="X83B9EFB47CE0A617" name="X83B9EFB47CE0A617"></a></p>
<h5>3.3-5 GroupForTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupForTom</code>( <var class="Arg">tomidentifier</var>[, <var class="Arg">repnr</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tomidentifier</var> be a string that is an admissible name for a table of marks from the <strong class="pkg">GAP</strong> Library of Tables of Marks (the <strong class="pkg">TomLib</strong> package <a href="chapBib.html#biBTomLib">[MNP19]</a>). Called with one argument, <code class="func">GroupForTom</code> returns the <code class="func">UnderlyingGroup</code> (<a href="../../../doc/ref/chap70.html#X81E41D3880FA6C4C"><span class="RefLink">Reference: UnderlyingGroup for tables of marks</span></a>) value of this table of marks. If a positive integer <var class="Arg">repnr</var> is given as the second argument then a representative of the <var class="Arg">repnr</var>-th class of subgroups of this group is returned, see <code class="func">RepresentativeTom</code> (<a href="../../../doc/ref/chap70.html#X7F625AB880B73AC3"><span class="RefLink">Reference: RepresentativeTom</span></a>).</p>
<p>The string<code class="code">"GroupForTom"</code> may occur in the entries of the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>), and therefore may be called by <code class="func">GroupForGroupInfo</code> (<a href="chap3.html#X804D2AC37E0D4755"><span class="RefLink">3.3-4</span></a>).</p>
<p>If the <strong class="pkg">TomLib</strong> package is not loaded or if it does not contain a table of marks with identifier <var class="Arg">tomidentifier</var> then <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GroupForTom( "A5" ); u:= GroupForTom( "A5", 2 );</span>
Group([ (2,4)(3,5), (1,2,5) ])
Group([ (2,3)(4,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset( g, u );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupForTom( "J4" );</span>
fail
</pre></div>
<p><a id="X8032EF83823B1D88" name="X8032EF83823B1D88"></a></p>
<h5>3.3-6 AtlasStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasStabilizer</code>( <var class="Arg">gapname</var>, <var class="Arg">repname</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">gapname</var> be an admissible name of a group <span class="SimpleMath">G</span>, say, in the sense of the <strong class="pkg">AtlasRep</strong> package (see Section <a href="../../../pkg/atlasrep/doc/chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">AtlasRep: Group Names Used in the AtlasRep Package</span></a>), and <var class="Arg">repname</var> be a string that occurs as the <code class="code">repname</code> component of a record returned by <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="../../../pkg/atlasrep/doc/chap3.html#X84C2D76482E60E42"><span class="RefLink">AtlasRep: AllAtlasGeneratingSetInfos</span></a>) when this function is called with first argument <var class="Arg">gapname</var> and further arguments <code class="func">IsTransitive</code> (<a href="../../../doc/ref/chap41.html#X79B15750851828CB"><span class="RefLink">Reference: IsTransitive</span></a>) and <code class="keyw">true</code>. In this case, <var class="Arg">repname</var> describes a transitive permutation representation of <span class="SimpleMath">G</span>.</p>
<p>If the <strong class="pkg">AtlasRep</strong> package is available and if the permutation group in question can be fetched then <code class="func">AtlasStabilizer</code> returns a point stabilizer. Otherwise <code class="keyw">fail</code> is returned.</p>
<p>The string<code class="code">"AtlasStabilizer"</code> may occur in the entries of the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>), and therefore may be called by <code class="func">GroupForGroupInfo</code> (<a href="chap3.html#X804D2AC37E0D4755"><span class="RefLink">3.3-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasStabilizer( "A5","A5G1-p5B0");</span>
Group([ (1,2)(3,4), (2,3,4) ])
</pre></div>
<p><a id="X7AADD757809DFDD5" name="X7AADD757809DFDD5"></a></p>
<h5>3.3-7 IsNontrivialDirectProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNontrivialDirectProduct</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var> of the group <span class="SimpleMath">G</span>, say, this function returns <code class="keyw">true</code> if <span class="SimpleMath">G</span> is the direct product of smaller groups, and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mx:= Maxes( CharacterTable( "J1" ) );</span>
[ "L2(11)", "2^3.7.3", "2xA5", "19:6", "11:10", "D6xD10", "7:6" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( mx, name -> IsNontrivialDirectProduct(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CharacterTable( name ) ) );</span>
[ false, false, true, false, false, true, false ]
</pre></div>
<p><a id="X7C1F6C01852D7933" name="X7C1F6C01852D7933"></a></p>
<h4>3.4 <span class="Heading">Unipotent Characters of Finite Groups of Lie Type</span></h4>
<p>Unipotent characters are defined for finite groups of Lie type. For most of these groups whose character table is in the <strong class="pkg">GAP</strong> Character Table Library, the unipotent characters are known and parametrised by labels. This labeling is due to the work of P. Deligne and G. Lusztig, thus the label of a unipotent character is called its Deligne-Lusztig name (see <a href="chapBib.html#biBCla05">[Cla05]</a>).</p>
<p><a id="X79F645F278C27F23" name="X79F645F278C27F23"></a></p>
<h5>3.4-1 UnipotentCharacter</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnipotentCharacter</code>( <var class="Arg">tbl</var>, <var class="Arg">label</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be the ordinary character table of a finite group of Lie type in the <strong class="pkg">GAP</strong> Character Table Library. <code class="func">UnipotentCharacter</code> returns the unipotent character with Deligne-Lusztig name <var class="Arg">label</var>.</p>
<p>The object <var class="Arg">label</var> must be either a list of integers which describes a partition (if the finite group of Lie type is of the type <span class="SimpleMath">A_l</span> or <span class="SimpleMath">^2A_l</span>), a list of two lists of integers which describes a symbol (if the group is of classical type other than <span class="SimpleMath">A_l</span> and <span class="SimpleMath">^2A_l</span>) or a string (if the group is of exceptional type).</p>
<p>A call of <code class="func">UnipotentCharacter</code> sets the attribute <code class="func">DeligneLusztigNames</code> (<a href="chap3.html#X803FD7F87AFBBDE4"><span class="RefLink">3.4-2</span></a>) for <var class="Arg">tbl</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "U4(2).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnipotentCharacter( tbl, [ [ 0, 1 ], [ 2 ] ] );</span>
Character( CharacterTable( "U4(2).2" ),
[ 15, 7, 3, -3, 0, 3, -1, 1, 0, 1, -2, 1, 0, 0, -1, 5, 1, 3, -1, 2,
-1, 1, -1, 0, 0 ] )
</pre></div>
<p><a id="X803FD7F87AFBBDE4" name="X803FD7F87AFBBDE4"></a></p>
<h5>3.4-2 DeligneLusztigNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeligneLusztigNames</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a character table <var class="Arg">obj</var>, <code class="func">DeligneLusztigNames</code> returns a list of Deligne-Lusztig names of the the unipotent characters of <var class="Arg">obj</var>. If the <span class="SimpleMath">i</span>-th entry is bound then it is the name of the <span class="SimpleMath">i</span>-th irreducible character of <var class="Arg">obj</var>, and this character is irreducible. If an irreducible character is not unipotent the accordant position is unbound.</p>
<p><code class="func">DeligneLusztigNames</code> called with a string <var class="Arg">obj</var>, calls itself with the argument <code class="code">CharacterTable( <var class="Arg">obj</var> )</code>.</p>
<p>When <code class="func">DeligneLusztigNames</code> is called with a record <var class="Arg">obj</var> then this should have the components <code class="code">isoc</code>, <code class="code">isot</code>, <code class="code">l</code>, and <code class="code">q</code>, where <code class="code">isoc</code> and <code class="code">isot</code> are strings defining the isogeny class and isogeny type, and <code class="code">l</code> and <code class="code">q</code> are integers. These components define a finite group of Lie type uniquely. Moreover this way one can choose Deligne-Lusztig names for a prescribed type in those cases where a group has more than one interpretation as a finite group of Lie type, see the example below. (The first call of <code class="func">DeligneLusztigNames</code> sets the attribute value in the character table.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeligneLusztigNames( "L2(7)" );</span>
[ [ 2 ],,,, [ 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L2(7)" );</span>
CharacterTable( "L3(2)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">HasDeligneLusztigNames( tbl );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DeligneLusztigNames( rec( isoc:= "A", isot:= "simple",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> l:= 2, q:= 2 ) );</span>
[ [ 3 ],,, [ 2, 1 ],, [ 1, 1, 1 ] ]
</pre></div>
<p><a id="X84B9D8EF84EB4145" name="X84B9D8EF84EB4145"></a></p>
<h5>3.4-3 DeligneLusztigName</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeligneLusztigName</code>( <var class="Arg">chi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a unipotent character <var class="Arg">chi</var>, <code class="func">DeligneLusztigName</code> returns the Deligne-Lusztig name of <var class="Arg">chi</var>. For that, <code class="func">DeligneLusztigNames</code> (<a href="chap3.html#X803FD7F87AFBBDE4"><span class="RefLink">3.4-2</span></a>) is called with the argument <code class="code">UnderlyingCharacterTable( <var class="Arg">chi</var> )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "F4(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeligneLusztigName( Irr( tbl )[9] );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">HasDeligneLusztigNames( tbl );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List( [ 1 .. 8 ], i -> DeligneLusztigName( Irr( tbl )[i] ) );</span>
[ "phi{1,0}", "[ [ 2 ], [ ] ]", "phi{2,4}''", "phi{2,4}'",
"F4^II[1]", "phi{4,1}", "F4^I[1]", "phi{9,2}" ]
</pre></div>
<p><a id="X78A05C0A82E23048" name="X78A05C0A82E23048"></a></p>
<h5>3.4-4 KnowsDeligneLusztigNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KnowsDeligneLusztigNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var>, this function returns <code class="keyw">true</code> if <code class="func">DeligneLusztigNames</code> (<a href="chap3.html#X803FD7F87AFBBDE4"><span class="RefLink">3.4-2</span></a>) returns the list of Deligne-Lusztig names of the unipotent characters of <var class="Arg">tbl</var>, and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">KnowsDeligneLusztigNames( CharacterTable( "A5" ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">KnowsDeligneLusztigNames( CharacterTable( "M" ) );</span>
false
</pre></div>
<p><a id="X7900893987A89915" name="X7900893987A89915"></a></p>
<h4>3.5 <span class="Heading"><strong class="pkg">Browse</strong> Applications Provided by
<strong class="pkg">CTblLib</strong></span></h4>
<p>The following functions are available only if the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse">[BL23]</a>) is loaded. The function <code class="func">DisplayCTblLibInfo</code> (<a href="chap3.html#X8684C73E844E7033"><span class="RefLink">3.5-1</span></a>) shows details about an ordinary or modular character table in a pager, the other functions can be used to show the following information via browse tables.</p>
<ul>
<li><p>An overview of the <strong class="pkg">GAP</strong> Character Table Library, and details pages about ordinary and modular character tables (see <code class="func">BrowseCTblLibInfo</code> (<a href="chap3.html#X7A038A267CD17032"><span class="RefLink">3.5-2</span></a>)), which allow one to navigate to related pages and to pages showing for example decomposition matrices (cf. <code class="func">BrowseDecompositionMatrix</code> (<a href="../../../pkg/browse/doc/chap6.html#X7D80A56D853C9655"><span class="RefLink">Browse: BrowseDecompositionMatrix</span></a>)),</p>
</li>
<li><p>an alternative display function that shows character tables from the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> in a format similar to the one used in these books (see <code class="func">BrowseAtlasTable</code> (<a href="chap3.html#X79DC72707B08A701"><span class="RefLink">3.5-9</span></a>), cf. <code class="func">Browse (for character tables)</code> (<a href="../../../pkg/browse/doc/chap6.html#X870C744182073CF6"><span class="RefLink">Browse: Browse for character tables</span></a>) for the default display format for character tables),</p>
</li>
<li><p>an overview of the names of simple groups for which the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> show the character tables and other information (see <code class="func">BrowseAtlasContents</code> (<a href="chap3.html#X8135745381E1C0E2"><span class="RefLink">3.5-5</span></a>), a variant that doe not rely on <strong class="pkg">Browse</strong> is <code class="func">DisplayAtlasContents</code> (<a href="chap3.html#X810789F77F20C839"><span class="RefLink">3.5-6</span></a>)),</p>
</li>
<li><p>a function that shows the <strong class="pkg">Atlas</strong> map of the bicyclic extensions of a simple <strong class="pkg">Atlas</strong> group (see <code class="func">BrowseAtlasMap</code> (<a href="chap3.html#X86AC6408815EF122"><span class="RefLink">3.5-7</span></a>), a variant that does not rely on <strong class="pkg">Browse</strong> is <code class="func">DisplayAtlasMap</code> (<a href="chap3.html#X875A6BB485A49976"><span class="RefLink">3.5-8</span></a>)),</p>
</li>
<li><p>an overview of the <q>atomic irrationalities</q> that occur in <strong class="pkg">Atlas</strong> character tables (see <code class="func">BrowseCommonIrrationalities</code> (<a href="chap3.html#X83FDD44E7E0AE885"><span class="RefLink">3.5-3</span></a>)),</p>
</li>
<li><p>an overview of the lists of improvements to the <strong class="pkg">Atlas</strong> of Finite Groups (see <code class="func">BrowseAtlasImprovements</code> (<a href="chap3.html#X79E3F783818FE75A"><span class="RefLink">3.5-10</span></a>)).</p>
</li>
<li><p>an overview of the differences between the character table data since version 1.1.3 of the <strong class="pkg">CTblLib</strong> package (see <code class="func">BrowseCTblLibDifferences</code> (<a href="chap3.html#X87CA74CF7B533CC6"><span class="RefLink">3.5-4</span></a>)),</p>
</li>
</ul>
<p>The functions <code class="func">BrowseCTblLibInfo</code> (<a href="chap3.html#X7A038A267CD17032"><span class="RefLink">3.5-2</span></a>), <code class="func">BrowseCommonIrrationalities</code> (<a href="chap3.html#X83FDD44E7E0AE885"><span class="RefLink">3.5-3</span></a>), <code class="func">BrowseCTblLibDifferences</code> (<a href="chap3.html#X87CA74CF7B533CC6"><span class="RefLink">3.5-4</span></a>), <code class="func">BrowseAtlasContents</code> (<a href="chap3.html#X8135745381E1C0E2"><span class="RefLink">3.5-5</span></a>), and <code class="func">BrowseAtlasImprovements</code> (<a href="chap3.html#X79E3F783818FE75A"><span class="RefLink">3.5-10</span></a>) occur also in the list of choices shown by <code class="func">BrowseGapData</code> (<a href="../../../pkg/browse/doc/chap6.html#X850C786C87A4877B"><span class="RefLink">Browse: BrowseGapData</span></a>).</p>
<p><a id="X8684C73E844E7033" name="X8684C73E844E7033"></a></p>
<h5>3.5-1 DisplayCTblLibInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayCTblLibInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayCTblLibInfo</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringCTblLibInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringCTblLibInfo</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>When <code class="func">DisplayCTblLibInfo</code> is called with an ordinary or modular character table <var class="Arg">tbl</var> then an overview of the information available for this character table is shown via the function that is given by the user preference <a href="chap4.html#X7E859C3482F27089"><span class="RefLink">4.5-3</span></a>. When <code class="func">DisplayCTblLibInfo</code> is called with a string <var class="Arg">name</var> that is an admissible name for an ordinary character table then the overview for this character table is shown. If a prime integer <var class="Arg">p</var> is entered in addition to <var class="Arg">name</var> then information about the <var class="Arg">p</var>-modular character table is shown instead.</p>
<p>An interactive variant of <code class="func">DisplayCTblLibInfo</code> is <code class="func">BrowseCTblLibInfo</code> (<a href="chap3.html#X7A038A267CD17032"><span class="RefLink">3.5-2</span></a>).</p>
<p>The string that is shown by <code class="func">DisplayCTblLibInfo</code> can be computed using <code class="func">StringCTblLibInfo</code>, with the same arguments.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">StringCTblLibInfo( CharacterTable( "A5" ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StringCTblLibInfo( CharacterTable( "A5" ) mod 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StringCTblLibInfo( "A5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StringCTblLibInfo( "A5", 2 );;</span>
</pre></div>
<p><a id="X7A038A267CD17032" name="X7A038A267CD17032"></a></p>
<h5>3.5-2 BrowseCTblLibInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseCTblLibInfo</code>( [<var class="Arg">func</var>, <var class="Arg">val</var>, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseCTblLibInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseCTblLibInfo</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p>Called without arguments, <code class="func">BrowseCTblLibInfo</code> shows the contents of the <strong class="pkg">GAP</strong> Character Table Library in an <em>overview table</em>, see below.</p>
<p>When arguments <var class="Arg">func</var>, <var class="Arg">val</var>, <var class="Arg">...</var> are given that are admissible arguments for <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>) –in particular, the first argument must be a function– then the overview is restricted to those character tables that match the conditions. The global option <code class="code">"OrderedBy"</code> is supported as in <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>).</p>
<p>When <code class="func">BrowseCTblLibInfo</code> is called with a character table <var class="Arg">tbl</var> then a <em>details table</em> is opened that gives an overview of the information available for this character table. When <code class="func">BrowseCTblLibInfo</code> is called with a string <var class="Arg">name</var> that is an admissible name for an ordinary character table then the details table for this character table is opened. If a prime integer <var class="Arg">p</var> is entered in addition to <var class="Arg">name</var> then information about the <var class="Arg">p</var>-modular character table is shown instead.</p>
<p>The overview table has the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the table,</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>the group order,</p>
</dd>
<dt><strong class="Mark"><code class="code">nccl</code></strong></dt>
<dd><p>the number of conjugacy classes,</p>
</dd>
<dt><strong class="Mark"><code class="code">fusions -> G</code></strong></dt>
<dd><p>the list of identifiers of tables on which a fusion to the given table is stored, and</p>
</dd>
<dt><strong class="Mark"><code class="code">fusions G -></code></strong></dt>
<dd><p>the list of identifiers of tables to which a fusion is stored on the given table.</p>
</dd>
</dl>
<p>The details table for a given character table has exactly one column. Only part of the functionality of the function <code class="func">NCurses.BrowseGeneric</code> (<a href="../../../pkg/browse/doc/chap4.html#X85FC163D87FAFD12"><span class="RefLink">Browse: NCurses.BrowseGeneric</span></a>) is available in such a table. On the other hand, the details tables contain <q>links</q> to other Browse applications, for example other details tables.</p>
<p>When one <q>clicks</q> on a row or an entry in the overview table then the details table for the character table in question is opened. One can navigate from this details table to a related one, by first <em>activating</em> a link (via repeatedly hitting the <strong class="button">Tab</strong> key) and then <em>following</em> the active link (via hitting the <strong class="button">Return</strong> key). If mouse actions are enabled (by hitting the <strong class="button">M</strong> key, see <code class="func">NCurses.UseMouse</code> (<a href="../../../pkg/browse/doc/chap2.html#X799C033A7AB582D7"><span class="RefLink">Browse: NCurses.UseMouse</span></a>)) then one can alternatively activate a link and click on it via mouse actions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tab:= [ 9 ];; # hit the TAB key</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= [ 14, 14, 14 ];; # ``do nothing'' input (means timeout)</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # select the first column, search for the name A5</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "sc/A5", [ NCurses.keys.DOWN, NCurses.keys.DOWN,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> NCurses.keys.RIGHT, NCurses.keys.ENTER ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # open the details table for A5</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ NCurses.keys.ENTER ], n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # activate the link to the character table of A5</span>
<span class="GAPprompt">></span> <span class="GAPinput"> tab, n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # show the character table of A5</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ NCurses.keys.ENTER ], n, n, "seddrr", n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # close this character table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # activate the link to the maximal subgroup D10</span>
<span class="GAPprompt">></span> <span class="GAPinput"> tab, tab, n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # jump to the details table for D10</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ NCurses.keys.ENTER ], n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # close this details table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # activate the link to a decomposition matrix</span>
<span class="GAPprompt">></span> <span class="GAPinput"> tab, tab, tab, tab, tab, n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # show the decomposition matrix</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ NCurses.keys.ENTER ], n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # close this table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # activate the link to the AtlasRep overview</span>
<span class="GAPprompt">></span> <span class="GAPinput"> tab, tab, tab, tab, tab, tab, tab, n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # show the overview</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ NCurses.keys.ENTER ], n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # close this table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # and quit the applications</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "QQ" ) );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseCTblLibInfo();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X83FDD44E7E0AE885" name="X83FDD44E7E0AE885"></a></p>
<h5>3.5-3 BrowseCommonIrrationalities</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseCommonIrrationalities</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of info records for the irrationalities that have been <q>clicked</q> in visual mode.</p>
<p>This function shows the atomic irrationalities that occur in character tables in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> or the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a>, together with descriptions of their reductions to the relevant finite fields in a browse table with the following columns. The format is the same as in <a href="chapBib.html#biBJLPW95">[JLPW95, Appendix 1]</a>.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the name of the irrationality, see <code class="func">AtlasIrrationality</code> (<a href="../../../doc/ref/chap18.html#X812E334E7A869D33"><span class="RefLink">Reference: AtlasIrrationality</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">p</code></strong></dt>
<dd><p>the characteristic,</p>
</dd>
<dt><strong class="Mark"><code class="code">value mod C_n</code></strong></dt>
<dd><p>the corresponding reduction to a finite field of characteristic <code class="code">p</code>, given by the residue modulo the <code class="code">n</code>-th Conway polynomial (see <code class="func">ConwayPolynomial</code> (<a href="../../../doc/ref/chap59.html#X7C2425A786F09054"><span class="RefLink">Reference: ConwayPolynomial</span></a>)),</p>
</dd>
<dt><strong class="Mark"><code class="code">n</code></strong></dt>
<dd><p>the degree of the smallest extension of the prime field of characteristic <code class="code">p</code> that contains the reduction.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= [ 14, 14, 14 ];; # ``do nothing'' input (means timeout)</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # categorize the table by the characteristics</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scrsc", n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # expand characteristic 2</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "srxq", n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # scroll down</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "DDD", n, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # and quit the application</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q" ) );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseCommonIrrationalities();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X87CA74CF7B533CC6" name="X87CA74CF7B533CC6"></a></p>
<h5>3.5-4 BrowseCTblLibDifferences</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseCTblLibDifferences</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p><code class="func">BrowseCTblLibDifferences</code> lists the differences between the versions of the character table data in the <strong class="pkg">CTblLib</strong> package, since version 1.1.3.</p>
<p>The overview table contains one row for each change, where <q>change</q> means the addition, modification, or removal of information, and has the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">Identifier</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the character table,</p>
</dd>
<dt><strong class="Mark"><code class="code">Type</code></strong></dt>
<dd><p>one of <code class="code">NEW</code> (for the addition of previously not available information), <code class="code">***</code> (for a bugfix), or <code class="code">C</code> (for a change that does not really fix a bug, typically a change motivated by a new consistency criterion),</p>
</dd>
<dt><strong class="Mark"><code class="code">What</code></strong></dt>
<dd><p>one of <code class="code">class fusions</code> (some class fusions from or to the table in question were changed), <code class="code">maxes</code> (the value of the attribute <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) was changed), <code class="code">names</code> (incorrect admissible names were removed), <code class="code">table</code> or <code class="code">table mod </code><span class="SimpleMath">p</span> (the ordinary or <span class="SimpleMath">p</span>-modular character table was changed), <code class="code">maxes</code> (the value of the attribute <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) was changed), <code class="code">tom fusion</code> (the value of the attribute <code class="func">FusionToTom</code> (<a href="chap3.html#X7B1AAED68753B1BE"><span class="RefLink">3.2-4</span></a>) was changed),</p>
</dd>
<dt><strong class="Mark"><code class="code">Description</code></strong></dt>
<dd><p>a description what has been changed,</p>
</dd>
<dt><strong class="Mark"><code class="code">Flag</code></strong></dt>
<dd><p>one of <code class="code">Dup</code> (the table is a duplicate, in the sense of <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>)), <code class="code">Der</code> (the row belongs to a character table that is derived from other tables), <code class="code">Fus</code> (the row belongs to the addition of class fusions), <code class="code">Max</code> (the row belongs to a character table that was added because its group is maximal in another group), or <code class="code">None</code> (in all other cases –these rows are to some extent the interesting ones). The information in this column can be used to restrict the overview to interesting subsets.</p>
</dd>
<dt><strong class="Mark"><code class="code">Vers.</code></strong></dt>
<dd><p>the package version in which the change described by the row appeared first.</p>
</dd>
</dl>
<p>The full functionality of the function <code class="func">NCurses.BrowseGeneric</code> (<a href="../../../pkg/browse/doc/chap4.html#X85FC163D87FAFD12"><span class="RefLink">Browse: NCurses.BrowseGeneric</span></a>) is available.</p>
<p>The following examples show the input for</p>
<ul>
<li><p>restricting the overview to error rows,</p>
</li>
<li><p>restricting the overview to <q>None</q> rows, and</p>
</li>
<li><p>restricting the overview to rows about a particular table.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= [ 14, 14, 14, 14, 14, 14 ];; # ``do nothing''</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">enter:= [ NCurses.keys.ENTER ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">down:= [ NCurses.keys.DOWN ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">right:= [ NCurses.keys.RIGHT ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scr", # select the 'Type' column,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "f***", enter, # filter rows containing '***',</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n, "Q" ) ); # and quit</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseCTblLibDifferences();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scrrrr", # select the 'Flag' column,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "fNone", enter, # filter rows containing 'None',</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n, "Q" ) ); # and quit</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseCTblLibDifferences();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "fM", # filter rows containing 'M',</span>
<span class="GAPprompt">></span> <span class="GAPinput"> down, down, down, right, # but 'M' as a whole word,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> enter, #</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n, "Q" ) ); # and quit</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseCTblLibDifferences();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X8135745381E1C0E2" name="X8135745381E1C0E2"></a></p>
<h5>3.5-5 BrowseAtlasContents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseAtlasContents</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p><code class="func">BrowseAtlasContents</code> shows the list of names of simple groups and the corresponding page numbers in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>, as given on page v of this book, plus a few groups for which <a href="chapBib.html#biBJLPW95">[JLPW95, Appendix 2]</a> states that their character tables in <strong class="pkg">Atlas</strong> format have been obtained; if applicable then also the corresponding page numbers in the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> are shown.</p>
<p>Clicking on a page number opens the <strong class="pkg">Atlas</strong> map for the group in question, see <code class="func">BrowseAtlasMap</code> (<a href="chap3.html#X86AC6408815EF122"><span class="RefLink">3.5-7</span></a>). (From the map, one can open the <strong class="pkg">Atlas</strong> style display using the input <code class="code">"T"</code>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= [ NCurses.keys.DOWN ];; r:= [ NCurses.keys.RIGHT ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= [ NCurses.keys.ENTER ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "/J2", # Find the string J2,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # start the search,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> r, # select the page for the ordinary table,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the entry,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "se", # select the box of the simple group,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the box,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the info overview for J2,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> d, # move down to 2.J2,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the box,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the info overview for 2.J2,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "T", # show the ATLAS table for (extensions of) J2</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the ATLAS table,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the map,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> r, # select the page for the 2-modular table,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the entry,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "T", # show the 2-modular ATLAS table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the ATLAS table,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the map,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q" ) ); # and quit the application.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseAtlasContents();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X810789F77F20C839" name="X810789F77F20C839"></a></p>
<h5>3.5-6 DisplayAtlasContents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasContents</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringAtlasContents</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DisplayAtlasContents</code> calls the function that is given by the user preference <a href="chap4.html#X7E859C3482F27089"><span class="RefLink">4.5-3</span></a>, in order to show the list of names of simple groups and the corresponding page numbers in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>, as given on page v of this book, plus a few groups for which <a href="chapBib.html#biBJLPW95">[JLPW95, Appendix 2]</a> states that their character tables in <strong class="pkg">Atlas</strong> format have been obtained; if applicable then also the corresponding page numbers in the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> are shown.</p>
<p>An interactive variant of <code class="func">DisplayAtlasContents</code> is <code class="func">BrowseAtlasContents</code> (<a href="chap3.html#X8135745381E1C0E2"><span class="RefLink">3.5-5</span></a>).</p>
<p>The string that is shown by <code class="func">DisplayAtlasContents</code> can be computed using <code class="func">StringAtlasContents</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">str:= StringAtlasContents();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= PositionNthOccurrence( str, '\n', 10 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( str{ [ 1 .. pos ] } );</span>
A5 = L2(4) = L2(5) 2 2:2, 3:2, 5:2
L3(2) = L2(7) 3 2:3, 3:3, 7:3
A6 = L2(9) = S4(2)' 4 2:4, 3:4, 5:5
L2(8) = R(3)' 6 2:6, 3:6, 7:6
L2(11) 7 2:7, 3:7, 5:8, 11:8
L2(13) 8 2:9, 3:9, 7:10, 13:10
L2(17) 9 2:11, 3:11, 17:12
A7 10 2:13, 3:13, 5:14, 7:15
L2(19) 11 2:16, 3:16, 5:17, 19:18
L2(16) 12 2:19, 3:20, 5:20, 17:21
</pre></div>
<p><a id="X86AC6408815EF122" name="X86AC6408815EF122"></a></p>
<h5>3.5-7 BrowseAtlasMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseAtlasMap</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p>For a string <var class="Arg">name</var> that is the identifier of the character table of a simple group from the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>, <code class="func">BrowseAtlasMap</code> shows the map that describes the bicyclic extensions of this group, see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 6]</a>. If the optional argument <var class="Arg">p</var> is not given or if <var class="Arg">p</var> is zero then the map for the ordinary character tables is shown, if <var class="Arg">p</var> is a prime integer then the map for the <var class="Arg">p</var>-modular Brauer character tables is shown, as in <a href="chapBib.html#biBJLPW95">[JLPW95]</a>.</p>
<p>Clicking on a square of the map opens the character table information for the extension in question, by calling <code class="func">BrowseCTblLibInfo</code> (<a href="chap3.html#X7A038A267CD17032"><span class="RefLink">3.5-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= [ NCurses.keys.DOWN ];; r:= [ NCurses.keys.RIGHT ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= [ NCurses.keys.ENTER ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "T", # show the ATLAS table for (extensions of) M12</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the ATLAS table,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "se", # select the box of the simple group,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the box,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the info overview for M12,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> r, d, # select the box for the bicyclic extension,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # click the box,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q", # quit the info overview,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q" ) ); # and quit the application.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseAtlasMap( "M12" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X875A6BB485A49976" name="X875A6BB485A49976"></a></p>
<h5>3.5-8 DisplayAtlasMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasMap</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasMap</code>( <var class="Arg">arec</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringsAtlasMap</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StringsAtlasMap</code>( <var class="Arg">arec</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="func">DisplayAtlasMap</code> returns nothing, <code class="func">StringsAtlasMap</code> returns either <code class="keyw">fail</code> or the list of strings that form the rows of the <strong class="pkg">Atlas</strong> map of the group in question.</p>
<p>Let <var class="Arg">name</var> be an admissible name for the character table of a simple <strong class="pkg">Atlas</strong> group, and <var class="Arg">p</var> be a prime integer or <span class="SimpleMath">0</span> (which is the default). <code class="func">DisplayAtlasMap</code> shows the map for the group and its extensions, similar to the map shown in the <strong class="pkg">Atlas</strong>. <code class="func">StringsAtlasMap</code> returns the list of strings that form the rows of this map.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( "M12" );</span>
--------- ---------
| | | |
| G | | G.2 | 15
| | | |
--------- ---------
--------- ---------
| | | |
| 2.G | | 2.G.2 | 11
| | | |
--------- ---------
15 9
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( "M12", 2 );</span>
--------- ---------
| | | |
| G | | G.2 | 6
| | | |
--------- ---------
6 0
<span class="GAPprompt">gap></span> <span class="GAPinput">StringsAtlasMap( "M11" );</span>
[ "--------- ", "| | ", "| G | 10", "| | ",
"--------- ", " 10 " ]
</pre></div>
<p>More generally, <var class="Arg">name</var> can be an admissible name for a character with known <code class="func">ExtensionInfoCharacterTable</code> (<a href="chap3.html#X82A008987DB887C2"><span class="RefLink">3.7-3</span></a>) value and such that the strings describing multiplier and outer automorphism group in this value occur in the lists <code class="code">CTblLib.AtlasMapMultNames</code> and <code class="code">CTblLib.AtlasMapOutNames</code>, respectively. If not all character tables of bicyclic extensions of the simple group in question are available then <code class="func">StringsAtlasMap</code> returns <code class="keyw">fail</code>, and <code class="func">DisplayAtlasMap</code> shows nothing.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( "S10(2)" );</span>
---------
| |
| G | 198
| |
---------
198
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( "L12(27)" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StringsAtlasMap( "L12(27)" );</span>
fail
</pre></div>
<p>If the abovementioned requirements are not satisfied for the character tables in question then one can provide the necessary information via a record <var class="Arg">arec</var>.</p>
<p>The following example shows the <q><strong class="pkg">Atlas</strong> map</q> for the alternating group on four points, viewed as an extension of the trivial group by a Klein four group and a group of order three.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( rec(</span>
<span class="GAPprompt">></span> <span class="GAPinput">labels:= [ [ "G", "G.3" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "2.G", "" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "2'.G", "" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "2''.G", "" ] ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">shapes:= [ [ "closed", "closed" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "closed", "empty" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "closed", "empty" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "closed", "empty" ] ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">labelscol:= [ "1", "1" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">labelsrow:= [ "1", "1", "1", "1" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">dashedhorz:= [ false, false, true, true ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">dashedvert:= [ false, false ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">showdashedrows:= true ) );</span>
--------- ---------
| | | |
| G | | G.3 | 1
| | | |
--------- ---------
---------
| |
| 2.G | 1
| |
---------
2'.G ---------
---------
2''.G ---------
---------
1 1
</pre></div>
<p>The next example shows the <q><strong class="pkg">Atlas</strong> map</q> for the symmetric group on three points, viewed as a bicyclic extension of the trivial group by groups of the orders three and two, respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasMap( rec(</span>
<span class="GAPprompt">></span> <span class="GAPinput">labels:= [ [ "G", "G.2" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "3.G", "3.G.2" ] ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">shapes:= [ [ "closed", "closed" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ "closed", "open" ] ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">labelscol:= [ "1", "1" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">labelsrow:= [ "1", "1" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">dashedhorz:= [ false, false ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">dashedvert:= [ false, false ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">showdashedrows:= true ) );</span>
--------- ---------
| | | |
| G | | G.2 | 1
| | | |
--------- ---------
--------- --------
| | |
| 3.G | | 3.G.2 1
| | |
---------
1 1
</pre></div>
<p>(Depending on the terminal capabilities, the results may look nicer than the <q>ASCII only</q> graphics shown above.)</p>
<p>The following components of <var class="Arg">arec</var> are supported.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>a string, the name of the (simple) group;</p>
</dd>
<dt><strong class="Mark"><code class="code">char</code></strong></dt>
<dd><p>the characteristic, the default is <span class="SimpleMath">0</span>;</p>
</dd>
<dt><strong class="Mark"><code class="code">identifiers</code></strong></dt>
<dd><p>an <span class="SimpleMath">m</span> by <span class="SimpleMath">n</span> matrix whose entries are <code class="keyw">fail</code> or the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap70.html#X810E53597B5BB4F8"><span class="RefLink">Reference: Identifier for tables of marks</span></a>) values of the character tables of the extensions in question;</p>
</dd>
<dt><strong class="Mark"><code class="code">labels</code></strong></dt>
<dd><p>an <span class="SimpleMath">m</span> by <span class="SimpleMath">n</span> matrix whose entries are <code class="keyw">fail</code> or the strings that shall be used as the labels of the boxes;</p>
</dd>
<dt><strong class="Mark"><code class="code">shapes</code></strong></dt>
<dd><p>an <span class="SimpleMath">m</span> by <span class="SimpleMath">n</span> matrix whose entries are the strings <code class="code">"closed"</code>, <code class="code">"open"</code>, <code class="code">"broken"</code>, and <code class="code">"empty"</code>, describing the boxes that occur;</p>
</dd>
<dt><strong class="Mark"><code class="code">labelscol</code></strong></dt>
<dd><p>a list of length <span class="SimpleMath">n</span> that contains the labels to be shown below the last row of boxes, intended to show the numbers of classes in this column of boxes;</p>
</dd>
<dt><strong class="Mark"><code class="code">labelsrow</code></strong></dt>
<dd><p>a list of length <span class="SimpleMath">m</span> that contains the labels to be shown on the right of the last column of boxes, intended to show the numbers of characters in this row of boxes;</p>
</dd>
<dt><strong class="Mark"><code class="code">dashedhorz</code></strong></dt>
<dd><p>a list of length <span class="SimpleMath">m</span> with entries <code class="keyw">true</code> (the boxes in this row shall have small height) or <code class="keyw">false</code> (the boxes in this row shall have normal height);</p>
</dd>
<dt><strong class="Mark"><code class="code">dashedvert</code></strong></dt>
<dd><p>a list of length <span class="SimpleMath">n</span> with entries <code class="keyw">true</code> (the boxes in this column shall have small width) or <code class="keyw">false</code> (the boxes in this column shall have normal width);</p>
</dd>
<dt><strong class="Mark"><code class="code">showdashedrows</code></strong></dt>
<dd><p><code class="keyw">true</code> or <code class="keyw">false</code>, the default is to show rows of <q>dashed</q> boxes in the case of ordinary tables, and to omit them in the case of Brauer tables, as happens in the printed Atlases;</p>
</dd>
<dt><strong class="Mark"><code class="code">onlyasciiboxes</code></strong></dt>
<dd><p><code class="keyw">true</code> (show only ASCII characters when drawing the boxes) or <code class="keyw">false</code> (use line drawing characters), the default is the value returned by <code class="code">CTblLib.ShowOnlyASCII</code>;</p>
</dd>
<dt><strong class="Mark"><code class="code">onlyasciilabels</code></strong></dt>
<dd><p><code class="keyw">true</code> (show only ASCII characters in the labels inside the boxes) or <code class="keyw">false</code> (default, use subscripts if applicable); the default is the value returned by <code class="code">CTblLib.ShowOnlyASCII</code>;</p>
</dd>
<dt><strong class="Mark"><code class="code">specialshapes</code></strong></dt>
<dd><p>a list of length three that describes exceptional cases (intended for the treatment of <q>dashed names</q> and <q>broken boxes</q>, look at the values in <code class="code">CTblLib.AtlasMapBoxesSpecial</code> where this component is actually used).</p>
</dd>
</dl>
<p><a id="X79DC72707B08A701" name="X79DC72707B08A701"></a></p>
<h5>3.5-9 BrowseAtlasTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseAtlasTable</code>( <var class="Arg">name</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p><code class="func">BrowseAtlasTable</code> displays the character tables of bicyclic extensions of the simple group with the name <var class="Arg">name</var> in a window, in the same format as the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> do. For that, it is necessary that these tables are known, as well as the class fusions between them and perhaps additional information (e. g., about the existence of certain extensions). These requirements are fulfilled if the tables are contained in the <strong class="pkg">Atlas</strong>, but they may hold also in other cases.</p>
<p>If a prime <var class="Arg">p</var> is given as the second argument then the <var class="Arg">p</var>-modular Brauer tables are shown, otherwise (or if <var class="Arg">p</var> is zero) the ordinary tables are shown.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= [ NCurses.keys.DOWN ];; r:= [ NCurses.keys.RIGHT ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= [ NCurses.keys.ENTER ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "/y", # Find the string y,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c, # start the search,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "nnnn", # Find more occurrences,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "Q" ) ); # and quit the application.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseAtlasTable( "A6" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p>The function uses <code class="func">NCurses.BrowseGeneric</code> (<a href="../../../pkg/browse/doc/chap4.html#X85FC163D87FAFD12"><span class="RefLink">Browse: NCurses.BrowseGeneric</span></a>). The identifier of the table is used as the static header. The strings <code class="code">X_1</code>, <code class="code">X_2</code>, <span class="SimpleMath">...</span> are used as row labels for those table rows that contain character values, and column labels are given by centralizer orders, power map information, and class names.</p>
<p><a id="X79E3F783818FE75A" name="X79E3F783818FE75A"></a></p>
<h5>3.5-10 BrowseAtlasImprovements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseAtlasImprovements</code>( [<var class="Arg">choice</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: nothing.</p>
<p>Called without argument or with the string <code class="code">"ordinary"</code>, <code class="func">BrowseAtlasImprovements</code> shows the lists of improvements to the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> that are contained in <a href="chapBib.html#biBBN95">[BN95]</a> and <a href="chapBib.html#biBAtlasImpII">[Nor]</a>.</p>
<p>Called with the string <code class="code">"modular"</code>, <code class="func">BrowseAtlasImprovements</code> shows the list of improvements to the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> that are contained in <a href="chapBib.html#biBABCImp">[ABC]</a>.</p>
<p>Called with <code class="keyw">true</code>, the concatenation of the above lists are shown.</p>
<p>The overview table contains one row for each improvement, and has the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">Section</code></strong></dt>
<dd><p>the part in the <strong class="pkg">Atlas</strong> to which the entry belongs (Introduction, The Groups, Additional information, Bibliography, Appendix 1, Appendix 2),</p>
</dd>
<dt><strong class="Mark"><code class="code">Src</code></strong></dt>
<dd><p><code class="code">1</code> for entries from <a href="chapBib.html#biBBN95">[BN95]</a>, <code class="code">2</code> for entries from <a href="chapBib.html#biBAtlasImpII">[Nor]</a>, and <code class="code">3</code> for entries from <a href="chapBib.html#biBABCImp">[ABC]</a>,</p>
</dd>
<dt><strong class="Mark"><code class="code">Typ</code></strong></dt>
<dd><p>the type of the improvement, one of <code class="code">***</code> (for mathematical errors), <code class="code">NEW</code> (for new information), <code class="code">C</code> (for improvements concerning grammar or notational consistency), or <code class="code">M</code> (for misprints or cases of illegibility),</p>
</dd>
<dt><strong class="Mark"><code class="code">Page</code></strong></dt>
<dd><p>the page and perhaps the line in the (ordinary or modular) <strong class="pkg">Atlas</strong>,</p>
</dd>
<dt><strong class="Mark"><code class="code">Group</code></strong></dt>
<dd><p>the name of the simple group to which the entry belongs (empty for entries not from the section <q>The Groups</q>),</p>
</dd>
<dt><strong class="Mark"><code class="code">Text</code></strong></dt>
<dd><p>the description of the entry,</p>
</dd>
<dt><strong class="Mark"><code class="code">**</code></strong></dt>
<dd><p>for each entry of the type <code class="code">***</code>, the subtype of the error to which some statements in <a href="chapBib.html#biBBMO17">[BMO17]</a> refer, one of <code class="code">CH</code> (character values), <code class="code">P</code> (power maps, element orders, and class names), <code class="code">FI</code> (fusions and indicators), <code class="code">I</code> (Introduction, Bibliography, the list showing the orders of multipliers and outer automorphism group, and the list of Conway polynomials), <code class="code">MP</code> (maps), <code class="code">MX</code> (descriptions of maximal subgroups), and <code class="code">G</code> (other information about the group).</p>
</dd>
</dl>
<p>The full functionality of the function <code class="func">NCurses.BrowseGeneric</code> (<a href="../../../pkg/browse/doc/chap4.html#X85FC163D87FAFD12"><span class="RefLink">Browse: NCurses.BrowseGeneric</span></a>) is available.</p>
<p>The following example shows the input for first restricting the list to errors (type <code class="code">***</code>), then categorizing the filtered list by the subtype of the error, and then expanding the category for the subtype <code class="code">CH</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= [ 14, 14, 14, 14, 14, 14 ];; # ``do nothing''</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">enter:= [ NCurses.keys.ENTER ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scrr", # select the 'Typ' column,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "f***", enter, # filter rows containing '***',</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scrrrrrrsc", enter, # categorize by the error kind</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "sr", enter, # expand the 'CH' category</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n, "Q" ) ); # and quit</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseAtlasImprovements();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BrowseData.SetReplay( false );</span>
</pre></div>
<p><a id="X7E5CFB187E313B11" name="X7E5CFB187E313B11"></a></p>
<h4>3.6 <span class="Heading">Duplicates of Library Character Tables</span></h4>
<p>It can be useful to deal with different instances of <q>the same</q> character table. An example is the situation that a group <span class="SimpleMath">G</span>, say, contains several classes of isomorphic maximal subgroups that have different class fusions; the attribute <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) of the character table of <span class="SimpleMath">G</span> then contains several entries that belong to the same group, but the identifiers of the character tables are different.</p>
<p>On the other hand, it can be useful to consider only one of the different instances when one searches for character tables with certain properties, for example using <code class="func">OneCharacterTableName</code> (<a href="chap3.html#X7EC4A57E8393D75C"><span class="RefLink">3.1-5</span></a>).</p>
<p>For that, we introduce the following concept. A character table <span class="SimpleMath">t_1</span> is said to be a <em>duplicate</em> of another character table <span class="SimpleMath">t_2</span> if the attribute <code class="func">IdentifierOfMainTable</code> (<a href="chap3.html#X860F49407882658F"><span class="RefLink">3.6-2</span></a>) returns the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of <span class="SimpleMath">t_2</span> when it is called with <span class="SimpleMath">t_1</span>, and we call <span class="SimpleMath">t_2</span> the <em>main table</em> of <span class="SimpleMath">t_1</span>. In this case, <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>) returns <code class="keyw">true</code> for <span class="SimpleMath">t_1</span>.</p>
<p>If the character table <span class="SimpleMath">t_1</span> is not a duplicate of any other library table then <code class="func">IdentifierOfMainTable</code> (<a href="chap3.html#X860F49407882658F"><span class="RefLink">3.6-2</span></a>) returns <code class="keyw">fail</code> for <span class="SimpleMath">t_1</span> and <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>) returns <code class="keyw">false</code>.</p>
<p>See <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>) for examples how to apply <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>) in practice.</p>
<p>We do <em>not</em> promise that two library tables for which <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>) returns <code class="keyw">false</code> are necessarily different. (And since nonisomorphic groups may have the same character table, it would not make sense to think about restricting a search to a subset of library tables that belong to pairwise nonisomorphic groups.)</p>
<p>Currently <code class="func">IdentifierOfMainTable</code> (<a href="chap3.html#X860F49407882658F"><span class="RefLink">3.6-2</span></a>) does not return <code class="keyw">fail</code> for <span class="SimpleMath">t_1</span> if <code class="func">ConstructionInfoCharacterTable</code> (<a href="chap3.html#X851118377D1D6EC9"><span class="RefLink">3.7-4</span></a>) is set in <span class="SimpleMath">t_1</span>, the first entry of the attribute value is <code class="code">"ConstructPermuted"</code>, and one of the following holds.</p>
<ul>
<li><p>The second entry of the <code class="func">ConstructionInfoCharacterTable</code> (<a href="chap3.html#X851118377D1D6EC9"><span class="RefLink">3.7-4</span></a>) value is a list of length <span class="SimpleMath">1</span> that contains the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of <span class="SimpleMath">t_2</span>.</p>
</li>
<li><p>The <strong class="pkg">SpinSym</strong> package is loaded and <span class="SimpleMath">t_1</span> is one of the character tables provided by this package. These tables are not declared as permuted tables of library tables, but we <em>want</em> to regard them as duplicates.</p>
</li>
</ul>
<p><a id="X7FD4A9DF7CBAA418" name="X7FD4A9DF7CBAA418"></a></p>
<h5>3.6-1 IsDuplicateTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDuplicateTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var> from the <strong class="pkg">GAP</strong> Character Table Library, this function returns <code class="keyw">true</code> if <var class="Arg">tbl</var> was constructed from another library character table by permuting rows and columns, via the attribute <code class="func">ConstructionInfoCharacterTable</code> (<a href="chap3.html#X851118377D1D6EC9"><span class="RefLink">3.7-4</span></a>). Otherwise <code class="keyw">false</code> is returned, in particular if <var class="Arg">tbl</var> is not a character table from the <strong class="pkg">GAP</strong> Character Table Library.</p>
<p>One application of this function is to restrict the search with <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>) to only one library character table for each class of permutation equivalent tables. Note that this property of the search result cannot be guaranteed if private character tables have been added to the library, see <code class="func">NotifyCharacterTable</code> (<a href="chap4.html#X79366F797CD02DAF"><span class="RefLink">4.7-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Maxes( CharacterTable( "A6" ) );</span>
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDuplicateTable( CharacterTable( "A5" ) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDuplicateTable( CharacterTable( "A6M2" ) );</span>
true
</pre></div>
<p><a id="X860F49407882658F" name="X860F49407882658F"></a></p>
<h5>3.6-2 IdentifierOfMainTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentifierOfMainTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">tbl</var> is an ordinary character table that is a duplicate in the sense of the introduction to Section <a href="chap3.html#X7E5CFB187E313B11"><span class="RefLink">3.6</span></a> then this function returns the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the main table of <var class="Arg">tbl</var>. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Maxes( CharacterTable( "A6" ) );</span>
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentifierOfMainTable( CharacterTable( "A5" ) );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentifierOfMainTable( CharacterTable( "A6M2" ) );</span>
"A5"
</pre></div>
<p><a id="X7C83435B78DE62F6" name="X7C83435B78DE62F6"></a></p>
<h5>3.6-3 IdentifiersOfDuplicateTables</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentifiersOfDuplicateTables</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var>, this function returns the list of <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) values of those character tables from the <strong class="pkg">GAP</strong> Character Table Library that are duplicates of <var class="Arg">tbl</var>, in the sense of the introduction to Section <a href="chap3.html#X7E5CFB187E313B11"><span class="RefLink">3.6</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Maxes( CharacterTable( "A6" ) );</span>
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentifiersOfDuplicateTables( CharacterTable( "A5" ) );</span>
[ "A6M2", "Alt(5)" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentifiersOfDuplicateTables( CharacterTable( "A6M2" ) );</span>
[ ]
</pre></div>
<p><a id="X7CCFEC998135F6FA" name="X7CCFEC998135F6FA"></a></p>
<h4>3.7 <span class="Heading">Attributes for Library Character Tables</span></h4>
<p>This section describes certain attributes which are set only for certain (not necessarily all) character tables from the <strong class="pkg">GAP</strong> Character Table Library. The attribute values are part of the database, there are no methods for <em>computing</em> them, except for <code class="func">InfoText</code> (<a href="chap3.html#X871562FD7F982C12"><span class="RefLink">3.7-5</span></a>) and the related property <code class="func">IsAtlasCharacterTable</code> (<a href="chap3.html#X867813EB79DC6953"><span class="RefLink">3.7-6</span></a>).</p>
<p>Other such attributes and properties are described in other manual sections because the context fits better. These attributes are <code class="func">FusionToTom</code> (<a href="chap3.html#X7B1AAED68753B1BE"><span class="RefLink">3.2-4</span></a>), <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>), <code class="func">KnowsSomeGroupInfo</code> (<a href="chap3.html#X87CFDC8081767CA4"><span class="RefLink">3.3-2</span></a>), <code class="func">IsNontrivialDirectProduct</code> (<a href="chap3.html#X7AADD757809DFDD5"><span class="RefLink">3.3-7</span></a>), <code class="func">DeligneLusztigNames</code> (<a href="chap3.html#X803FD7F87AFBBDE4"><span class="RefLink">3.4-2</span></a>), <code class="func">DeligneLusztigName</code> (<a href="chap3.html#X84B9D8EF84EB4145"><span class="RefLink">3.4-3</span></a>), <code class="func">KnowsDeligneLusztigNames</code> (<a href="chap3.html#X78A05C0A82E23048"><span class="RefLink">3.4-4</span></a>), <code class="func">IsDuplicateTable</code> (<a href="chap3.html#X7FD4A9DF7CBAA418"><span class="RefLink">3.6-1</span></a>), and <code class="func">CASInfo</code> (<a href="chap4.html#X786A80A279674E91"><span class="RefLink">4.4-1</span></a>).</p>
<p><a id="X8150E63F7DBDF252" name="X8150E63F7DBDF252"></a></p>
<h5>3.7-1 Maxes</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Maxes</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If this attribute is set for an ordinary character table <var class="Arg">tbl</var> then the value is a list of identifiers of the ordinary character tables of all maximal subgroups of <var class="Arg">tbl</var>. There is no default method to <em>compute</em> this value from <var class="Arg">tbl</var>.</p>
<p>If the <code class="func">Maxes</code> value of <var class="Arg">tbl</var> is stored then it lists exactly one representative for each conjugacy class of maximal subgroups of the group of <var class="Arg">tbl</var>, and the character tables of these maximal subgroups are available in the <strong class="pkg">GAP</strong> Character Table Library, and compatible class fusions to <var class="Arg">tbl</var> are stored on these tables (see the example in Section <a href="chap2.html#X83BF83D87BC1B123"><span class="RefLink">2.3-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "M11" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasMaxes( tbl );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">maxes:= Maxes( tbl );</span>
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( maxes[1] );</span>
CharacterTable( "A6.2_3" )
</pre></div>
<p><a id="X82DC2E7779322DA8" name="X82DC2E7779322DA8"></a></p>
<h5>3.7-2 ProjectivesInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectivesInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If this attribute is set for an ordinary character table <var class="Arg">tbl</var> then the value is a list of records, each with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the character table of the covering whose faithful irreducible characters are described by the record,</p>
</dd>
<dt><strong class="Mark"><code class="code">chars</code></strong></dt>
<dd><p>a list of values lists of faithful projective irreducibles; only one representative of each family of Galois conjugates is contained in this list. and</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectivesInfo( CharacterTable( "A5" ) );</span>
[ rec(
chars := [ [ 2, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ],
[ 2, 0, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ],
[ 4, 0, 1, -1, -1 ], [ 6, 0, 0, 1, 1 ] ], name := "2.A5" ) ]
</pre></div>
<p><a id="X82A008987DB887C2" name="X82A008987DB887C2"></a></p>
<h5>3.7-3 ExtensionInfoCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtensionInfoCharacterTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be the ordinary character table of a group <span class="SimpleMath">G</span>, say. If this attribute is set for <var class="Arg">tbl</var> then the value is a list of length two, the first entry being a string <code class="code">M</code> that describes the Schur multiplier of <span class="SimpleMath">G</span> and the second entry being a string <code class="code">A</code> that describes the outer automorphism group of <span class="SimpleMath">G</span>. Trivial multiplier or outer automorphism group are denoted by an empty string.</p>
<p>If <var class="Arg">tbl</var> is a table from the <strong class="pkg">GAP</strong> Character Table Library and <span class="SimpleMath">G</span> is (nonabelian and) simple then the value is set. In this case, an admissible name for the character table of a universal covering group of <span class="SimpleMath">G</span> (if this table is available and different from <var class="Arg">tbl</var>) is given by the concatenation of <code class="code">M</code>, <code class="code">"."</code>, and the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of <var class="Arg">tbl</var>. Analogously, an admissible name for the character table of the automorphism group of <span class="SimpleMath">G</span> (if this table is available and different from <var class="Arg">tbl</var>) is given by the concatenation of the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of <var class="Arg">tbl</var>, <code class="code">"."</code>, and <code class="code">A</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtensionInfoCharacterTable( CharacterTable( "A5" ) );</span>
[ "2", "2" ]
</pre></div>
<p><a id="X851118377D1D6EC9" name="X851118377D1D6EC9"></a></p>
<h5>3.7-4 ConstructionInfoCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstructionInfoCharacterTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If this attribute is set for an ordinary character table <var class="Arg">tbl</var> then the value is a list that describes how this table was constructed. The first entry is a string that is the identifier of the function that was applied to the pre-table record; the remaining entries are the arguments for that function, except that the pre-table record must be prepended to these arguments.</p>
<p><a id="X871562FD7F982C12" name="X871562FD7F982C12"></a></p>
<h5>3.7-5 InfoText</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoText</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>This method for library character tables returns an empty string if no <code class="func">InfoText</code> value is stored on the table <var class="Arg">tbl</var>.</p>
<p>Without this method, it would be impossible to use <code class="func">InfoText</code> in calls to <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>), as in the following example.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllCharacterTableNames( InfoText,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> s -> PositionSublist( s, "tests:" ) <> fail );;</span>
</pre></div>
<p><a id="X867813EB79DC6953" name="X867813EB79DC6953"></a></p>
<h5>3.7-6 IsAtlasCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAtlasCharacterTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>For an ordinary character table <var class="Arg">tbl</var>, this function returns <code class="keyw">true</code> if and only if the <code class="func">InfoText</code> (<a href="chap3.html#X871562FD7F982C12"><span class="RefLink">3.7-5</span></a>) value of <var class="Arg">tbl</var> contains the string <code class="code">"origin: ATLAS of finite groups"</code>.</p>
<p>For a Brauer character table <var class="Arg">tbl</var>, this function returns <code class="keyw">true</code> if and only if <code class="func">IsAtlasCharacterTable</code> returns <code class="keyw">true</code> for the underlying ordinary table of <var class="Arg">tbl</var>.</p>
<p>One application of this function is to restrict the search with <code class="func">AllCharacterTableNames</code> (<a href="chap3.html#X7C091641852BB6FE"><span class="RefLink">3.1-4</span></a>) to character tables from the <strong class="pkg">Atlas</strong> of Finite Groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Maxes( CharacterTable( "A6" ) );</span>
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAtlasCharacterTable( CharacterTable( "A5" ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAtlasCharacterTable( CharacterTable( "A5" ) mod 2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAtlasCharacterTable( CharacterTable( "A6M2" ) );</span>
false
</pre></div>
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