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"http://www.w3.org/TR/html4/loose.dtd"> <html> <meta name="GENERATOR" content="TtH 3.59"> <style type="text/css"> div.p { margin-top: 7pt;}</style> <style type="text/css"><!-- td div.comp { margin-top: -0.6ex; margin-bottom: -1ex;} td div.comb { margin-top: -0.6ex; margin-bottom: -.6ex;} td div.hrcomp { line-height: 0.9; margin-top: -0.8ex; margin-bottom: -1ex;} td div.norm {line-height:normal;} span.roman {font-family: serif; font-style: normal; font-weight: normal;} span.overacc2 {position: relative; left: .8em; top: -1.2ex;} span.overacc1 {position: relative; left: .6em; top: -1.2ex;} --> <title>Some steps in the verification of the ordinary character table of the Monster group</titl <h1 align="center">Some steps in the verification of the ordinary character table of the Monste <body bgcolor="FFFFFF"> <div class="p"><!----></div> <h3 align="center"> T<font size="-2">HOMAS</font> B<font size="-2">REUER</font>, K<font size="-2">A <div class="p"><!----></div> <h3 align="center">December 12th, 2024 </h3> <div class="p"><!----></div> <div class="p"><!----></div> We show the details of certain computations that are used in [<a href="#Mverify" name="CITEMve <div class="p"><!----></div> <div class="p"><!----></div> <h1>Contents </h1><a href="#tth_sEc1" >1 Overview</a><br /><a href="#tth_sEc2" >2 Some restrictions of the natural character of <font size="+0">M</font></a><br /><a href="#tth_sEc3 >3 The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup> >4 The conjugacy classes of <font size="+0">M</font></a><br /> <a href="#tth_sEc4.1" >4.1 Our strategy to describe the conjugacy classes of <font size="+0">M</font></a><br /> <a href="#tth >4.2 Utility functions</a><br /> <a href="#tth_sEc4.3" >4.3 Classes of elements of even order</a><br /> <a href="#tth_sEc4.4" >4.4 Classes of elements of order divisible by 3</a><br /> <a href="#tth_sEc4.5" >4.5 Classes of elements of order divisible by 5</a><br /> <a href="#tth_sEc4.6" >4.6 Classes of elements of order divisible by 11</a><br /> <a href="#tth_sEc4.7" >4.7 Classes of elements of the orders 17, 19, 23, 31, 47</a><br /> <a href="#tth_sEc4.8" >4.8 Classes of elements of order 13</a><br /> <a href="#tth_sEc4.9" >4.9 Classes of elements of order divisible by 29</a><br /> <a href="#tth_sEc4.10" >4.10 Classes of elements of order divisible by 41</a><br /> <a href="#tth_sEc4.11" >4.11 Classes of elements of order divisible by 59</a><br /> <a href="#tth_sEc4.12" >4.12 Classes of elements of order divisible by 71</a><br /> <a href="#tth_sEc4.13" >4.13 Classes of elements of order divisible by 7</a><br /><a href="#tth_sEc5" >5 The power maps of <font size="+0">M</font></a><br /><a href="#tth_sEc6" >6 The degree 196 883 character χ of <font size="+0">M</font></a><br /><a href="#tth_sEc7" >7 The irreducible characters of <font size="+0">M</font></a><br /><a href="#tth_sEc8" >8 Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</s >9 Appendix: The character table of 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2</ >9.1 Overview</a><br /> <a href="#tth_sEc9.2" >9.2 A permutation representation of H / X</a><br /> <a href="#tth_sEc9.3" >9.3 A permutation representation of H</a><br /> <a href="#tth_sEc9.4" >9.4 Compute the character table of H</a><br /><a href="#tth_sEc10" >10 Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</ <div class="p"><!----></div> <div class="p"><!----></div> <h2><a name="tth_sEc1"> 1</a> Overview</h2> <div class="p"><!----></div> The aim of [<a href="#Mverify" name="CITEMverify">BMW24</a>] is to verify the ordinary characte of the Monster group <font size="+0">M</font>. Here we collect, in the form of an explicit and reproducible <font face="helvetica">GAP</font> [<a href="#GAP" name="CITEGAP">GAP24</a>] session protocol, the relevant computations that are needed in that paper. <div class="p"><!----></div> We proceed as follows. <div class="p"><!----></div> Section <a href="#natural">2</a> verifies the decomposition of the restrictions of the ordinary irreducible character of degree 196 883 of <font size="+0">M</font> to the subgroups 2.<font size="+0">B</font> and 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> as stated in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 1]. <div class="p"><!----></div> Section <a href="#suborbits">3</a> verifies the decompositions of the transitive constituents of the permutation character of the action of C<sub><font size="+0">M</font></sub>(a) ≅ 2.<font size="+0">B</font> on the conjugacy cl where a is a <tt>2A</tt> involution in <font size="+0">M</font>. <div class="p"><!----></div> Sections <a href="#Mclasses">4</a> and <a href="#Mpowermaps">5</a> construct the character table head of <font size="+0">M</font>, that is, the lists of conjugacy class lengths, element orders, and power maps. <div class="p"><!----></div> Section <a href="#sect:natcharM">6</a> constructs the values of the irreducible degree 196 883 character of <font size="+0">M</font> and decides the isomorphism type of the <tt>3B</tt> normalizer in <font size="+0">M</font>. <div class="p"><!----></div> With this information and with the (already verified) character tables of the subgroups 2.<font size="+0">B</font>, 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</spa 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 of <font size="+0">M</font>, computing the irreducible characters of <font size="+0">M</font> is then easy; this corresponds to [<a href="#Mverify" name="CITEMverify">BMW24</a>,Section 5] and is done in Section <a href="#sect:irreduciblesM">7</a>. <div class="p"><!----></div> The final sections <a href="#sect:table_c2b">8</a>, <a href="#sect:norm3B">9</a>, <a href="#sect:table_N5B">10</a> document the constructions of thr character tables of subgroups of <font size="+0">M</font>. <div class="p"><!----></div> We will use the <font face="helvetica">GAP</font> Character Table Library and the interface to the A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" thus we load these <font face="helvetica">GAP</font> packages. <div class="p"><!----></div> <pre> gap> LoadPackage( "ctbllib", false ); true gap> LoadPackage( "atlasrep", false ); true </pre> <div class="p"><!----></div> The <font face="helvetica">MAGMA</font> system [<a href="#Magma" name="CITEMagma">BCP97</a>] wi for computing some character tables and for many conjugacy tests. If the following command returns <tt>false</tt> then these steps will not work. <div class="p"><!----></div> <pre> gap> CTblLib.IsMagmaAvailable(); true </pre> <div class="p"><!----></div> We set the line length to 72, like in other standard testfiles. <div class="p"><!----></div> <pre> gap> SizeScreen( [ 72 ] );; </pre> <div class="p"><!----></div> <h2><a name="tth_sEc2"> 2</a> Some restrictions of the natural character of <font size="+0">M</font></h2><a name="natural"> </a> <div class="p"><!----></div> We assume the existence of an ordinary irreducible character χ of degree 196 883 of the Monster group <font size="+0">M</font>, and that <font size="+0">M</font> has only two conjugacy classes of involutions. <div class="p"><!----></div> First we compute the restriction of χ to 2.<font size="+0">B</font>. <div class="p"><!----></div> The only faithful degree 196 883 character of 2.<font size="+0">B</font> that has at most two different values on involutions is 1a + 4371a + 96255a + 96256a, as claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lem This follows from the following data about 2.<font size="+0">B</font>. <div class="p"><!----></div> <pre> gap> table2B:= CharacterTable( "2.B" );; gap> cand:= Filtered( Irr( table2B ), x -> x[1] <= 196883 );; gap> List( cand, x -> x[1] ); [ 1, 4371, 96255, 96256 ] gap> inv:= Positions( OrdersClassRepresentatives( table2B ), 2 ); [ 2, 3, 4, 5, 7 ] gap> PrintArray( List( cand, x -> x{ Concatenation( [ 1 ], inv ) } ) ); [ [ 1, 1, 1, 1, 1, 1 ], [ 4371, 4371, -493, 275, 275, 19 ], [ 96255, 96255, 4863, 2047, 2047, 255 ], [ 96256, -96256, 0, 2048, -2048, 0 ] ] </pre> <div class="p"><!----></div> Note that 96256a must occur as a constituent because it is the only faithful candidate, and it can occur only once because otherwise only 4371a + 2 ·96256a or 4371 ·1a + 2 ·96256a would be possible decompositions, which have more than two values on involution classes. Thus the values of χ<sub>2.<font size="+0">B</font></sub> on the classes 4 and 5 differ by 4096. <div class="p"><!----></div> If 96255a would <b>not</b> occur then the values of χ<sub>2.<font size="+0">B</font></sub> on the classes 3 and 7 would differ by 512 times the multiplicity of 4371, but 65659 ·1a + 8 ·4371a + 96256a is not a solution. Thus 96255a must occur exactly once. <div class="p"><!----></div> The sum of 96255a and 96256a has four different values on involutions, hence also 4371a must occur. <div class="p"><!----></div> We see that the values of χ on the classes of involutions are 4371 and 275, respectively. <div class="p"><!----></div> <pre> gap> Sum( cand ){ inv }; [ 4371, 4371, 4371, 275, 275 ] </pre> <div class="p"><!----></div> The restriction of χ to 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> is computed similarly, as follows. <div class="p"><!----></div> Exactly seven irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> can constituents of the restriction of χ. <div class="p"><!----></div> <pre> gap> table3Fi24prime:= CharacterTable( "3.Fi24'" );; gap> cand:= Filtered( Irr( table3Fi24prime ), x -> x[1] <= 196883 );; gap> inv:= Positions( OrdersClassRepresentatives( table3Fi24prime ), 2 ); [ 4, 7 ] gap> mat:= List( cand, x -> x{ Concatenation([1], inv)});; gap> PrintArray( mat ); [ [ 1, 1, 1 ], [ 8671, 351, -33 ], [ 57477, 1157, 133 ], [ 783, 79, 15 ], [ 783, 79, 15 ], [ 64584, 1352, 72 ], [ 64584, 1352, 72 ] ] </pre> <div class="p"><!----></div> Since χ is rational, we need to consider only rationally irreducible characters, that is, the possible constituents are 1a, 8671a, 57377a, 783ab, and 64584ab. <div class="p"><!----></div> <pre> gap> List( cand, x -> x[2] ); [ 1, 8671, 57477, 783*E(3), 783*E(3)^2, 64584*E(3), 64584*E(3)^2 ] </pre> <div class="p"><!----></div> We see that the value on the first class of involutions must be 4371, since all values of the possible constituents are positive and too large for the other possible value 275. <div class="p"><!----></div> Since the values of all possible constituents on the second class of involutions are at most equal to the values on the first class, and equal only for 1a, we conclude that the value on the second class of involutions is 275. <div class="p"><!----></div> We see from the ratio of the value on the identity element and on the first class of involutions that constituents of degree 57477 or 2 ·64584 exist. <div class="p"><!----></div> <pre> gap> Float( 196883 / 4371 ); 45.043 gap> List( mat, v -> Float( v[1] / v[2] ) ); [ 1., 24.7037, 49.6776, 9.91139, 9.91139, 47.7692, 47.7692 ] gap> Float( ( 196883 - 2 * 64584 ) / ( 4371 - 2 * 1352 ) ); 40.6209 gap> Float( ( 196883 - 57477 ) / ( 4371 - 1157 ) ); 43.3746 gap> Float( ( 196883 - 2*57477 ) / ( 4371 - 2*1157 ) ); 39.8294 gap> Float( ( 196883 - 3*57477 ) / ( 4371 - 3*1157 ) ); 27.1689 </pre> <div class="p"><!----></div> First suppose that 64584ab is not a constituent. The above ratios imply that (at least) three constituents of degree 57477 must occur. <div class="p"><!----></div> However, then the degree admits at most two constituents of degree 8671, hence the value on the second class of involutions cannot be 275, a contradiction. <div class="p"><!----></div> This means that both 64584ab and 57477a occur with multiplicity one. <div class="p"><!----></div> <pre> gap> mat[3] + mat[6] + mat[7]; [ 186645, 3861, 277 ] </pre> <div class="p"><!----></div> The second involution class forces one constituent of degree 8671 (which is the only candidate that can contribute a negative value), and then a character of degree 1567 remains to be decomposed. The only solution for the degrees of its constituents is 1 + 1566. We get the decomposition 1a + 8671a + 57477a + 783ab + 64584ab, as claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 2]. <div class="p"><!----></div> <pre> gap> Sum( mat ); [ 196883, 4371, 275 ] </pre> <div class="p"><!----></div> The characters of the degrees 1, 8671, and 57477 extend two-fold from 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> to 3.<span class="roman">Fi</span><sub>24</ In order to decompose the restriction of χ to 3.<span class="roman">Fi</span><sub>24</sub>, we have to determine which extensions from 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> oc The following irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub> can occur as constituents of the restriction of χ. <div class="p"><!----></div> <pre> gap> table3Fi24:= CharacterTable( "3.Fi24" );; gap> cand:= Filtered( Irr( table3Fi24 ), x -> x[1] <= 196883 );; gap> inv:= Positions( OrdersClassRepresentatives( table3Fi24 ), 2 ); [ 3, 5, 172, 173 ] gap> mat:= List( cand, x -> x{ Concatenation([1], inv)});; gap> PrintArray( mat ); [ [ 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1 ], [ 8671, 351, -33, 1495, -41 ], [ 8671, 351, -33, -1495, 41 ], [ 57477, 1157, 133, 5865, 233 ], [ 57477, 1157, 133, -5865, -233 ], [ 1566, 158, 30, 0, 0 ], [ 129168, 2704, 144, 0, 0 ] ] </pre> <div class="p"><!----></div> We get the decomposition 1a + 8671b + 57477a + 1566a + 129168a claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a> <div class="p"><!----></div> <pre> gap> Sum( mat{ [ 1, 4, 5, 7, 8 ] } ); [ 196883, 4371, 275, 4371, 275 ] </pre> <div class="p"><!----></div> <h2><a name="tth_sEc3"> 3</a> The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></s </a> <div class="p"><!----></div> According to [<a href="#GMS89" name="CITEGMS89">GMS89</a>,Tables VII, IX], the restriction of the permutation character 1<sub>2.<font size="+0">B</font></sub><sup><font size= decomposes into nine transitive permutation characters 1<sub>U</sub><sup>2.<font size="+0">B</fo with the point stabilizers U listed in Table <a href="#suborbitsTable">1</a>. <div class="p"><!----></div> <div class="p"><!----></div> <a name="tth_tAb1"> </a> <center>Table 1: Suborbit information</center><a name="suborbitsTable"> </a> <center> <table> <tr><td>a c ∈ </td><td>G<sub>a,c</sub> </td><td align="right">|c<sup>G<sub>a</sub></sup>| </td></tr> <tr><td><tt>1A</tt> </td><td>2.<font size="+0">B</font> </td><td align="right">1 </td></tr> <tr><td><tt>2A</tt> </td><td>2<sup>2</sup>.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2) </td><td align=" <tr><td><tt>2B</tt> </td><td>2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> </td><td align="right">1 <tr><td><tt>3A</tt> </td><td><span class="roman">Fi</span><sub>23</sub> </td><td align="right">2031941058560 <tr><td><tt>3C</tt> </td><td><span class="roman">Th</span> </td><td align="right">91569524834304000 </td></t <tr><td><tt>4A</tt> </td><td>2<sup>1+22</sup>.<span class="roman">McL</span> </td><td align="right">11029353 <tr><td><tt>4B</tt> </td><td>2.<span class="roman">F</span><sub>4</sub>(2) </td><td align="right">1254793905 <tr><td><tt>5A</tt> </td><td><span class="roman">HN</span> </td><td align="right">30434513446055706624 </t <tr><td><tt>6A</tt> </td><td>2.<span class="roman">Fi</span><sub>22</sub> </td><td align="right">64353605265 </center> <div class="p"><!----></div> Here a denotes the central involution in 2.<font size="+0">B</font>, the action is that on the <font size="+0">M</font>-conjugacy class of a, and c ∈ a<sup><font size="+0">M</font></sup> is a representative of the orbit in question. <div class="p"><!----></div> In this section, we compute the nine characters 1<sub>U</sub><sup>2.<font size="+0">B</font></sup>, where U is one of the above point stabilizers G<sub>a,c</sub>. Note that a ∈ G<sub>a,c</sub> holds (and thus the character is an inflated character of <font size="+0">B</font>) if and only if a and c commute; this happens exactly for the first three orbits. <div class="p"><!----></div> All subgroups U except 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> and 2<sup>1+22</sup>.< are A<font size="-2">TLAS</font> groups whose character tables have been verified. The subgroup 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> is the preimage of a maximal s 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub> of <font size="+0">B</font> under the natura and the computation/verification of the character table of 2<sup>1+22</sup>.<span class="roman has been described in [<a href="#BMverify" name="CITEBMverify">BMW20</a>]. It will turn out that we do not need the character table of 2<sup>1+22</sup>.<span class="roman">McL< <div class="p"><!----></div> The nine characters will be stored in the variables <tt>pi1</tt>, <tt>pi2</tt>, ..., <tt>pi9</tt>. <div class="p"><!----></div> For U = 2.<font size="+0">B</font>, we have 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> = 1<sub>2.<font size="+0">B</font></sub>. <div class="p"><!----></div> <pre> gap> pi1:= TrivialCharacter( table2B );; </pre> <div class="p"><!----></div> For U = 2<sup>2</sup>.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2), the character 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> is the inflation of 1<sub>[U]</sub><sup><f from <font size="+0">B</font> to 2.<font size="+0">B</font>, for [U] = U / 〈a 〉 = 2.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2). (Note that the class fusion from [U] to <font size="+0">B</font> is not uniquely determined by the character tables of the two groups, but the permutation character is unique.) <div class="p"><!----></div> <pre> gap> tableB:= CharacterTable( "B" );; gap> tableUbar:= CharacterTable( "2.2E6(2)" );; gap> fus:= PossibleClassFusions( tableUbar, tableB );; gap> pi:= Set( fus, > map -> InducedClassFunctionsByFusionMap( tableUbar, tableB, > [ TrivialCharacter( tableUbar ) ], map )[1] );; gap> Length( pi ); 1 gap> pi2:= Inflated( tableB, table2B, pi )[1];; gap> mult:= List( Irr( table2B ), > chi -> ScalarProduct( table2B, chi, pi2 ) );; gap> Maximum( mult ); 1 gap> Positions( mult, 1 ); [ 1, 2, 3, 5, 7, 13, 15, 17 ] </pre> <div class="p"><!----></div> For U = 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>, the character 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> is the inflation of 1<sub>[U]</sub><sup><f from <font size="+0">B</font> to 2.<font size="+0">B</font>, for [U] = U / 〈a 〉 = 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub>, a maximal subgroup of <font size="+0">B</font>. <div class="p"><!----></div> <pre> gap> tableUbar:= CharacterTable( "BN2B" ); CharacterTable( "2^(1+22).Co2" ) gap> fus:= PossibleClassFusions( tableUbar, tableB );; gap> pi:= Set( fus, > map -> InducedClassFunctionsByFusionMap( tableUbar, tableB, > [ TrivialCharacter( tableUbar ) ], map )[1] );; gap> Length( pi ); 1 gap> pi3:= Inflated( tableB, table2B, pi )[1];; gap> mult:= List( Irr( table2B ), > chi -> ScalarProduct( table2B, chi, pi3 ) );; gap> Maximum( mult ); 1 gap> Positions( mult, 1 ); [ 1, 3, 5, 8, 13, 15, 28, 30, 37, 40 ] </pre> <div class="p"><!----></div> Next we consider U = <span class="roman">Fi</span><sub>23</sub>. <div class="p"><!----></div> <pre> gap> tableU:= CharacterTable( "Fi23" );; gap> fus:= PossibleClassFusions( tableU, table2B );; gap> pi:= Set( fus, > map -> InducedClassFunctionsByFusionMap( tableU, table2B, > [ TrivialCharacter( tableU ) ], map )[1] );; gap> Length( pi ); 1 gap> pi4:= pi[1];; gap> mult:= List( Irr( table2B ), > chi -> ScalarProduct( table2B, chi, pi4 ) );; gap> Maximum( mult ); 1 gap> Positions( mult, 1 ); [ 1, 2, 3, 5, 7, 8, 9, 12, 13, 15, 17, 23, 27, 30, 32, 40, 41, 54, 63, 68, 77, 81, 83, 185, 186, 187, 188, 189, 194, 195, 196, 203, 208, 220 ] </pre> <div class="p"><!----></div> Next we consider U = <span class="roman">Th</span>. <div class="p"><!----></div> <pre> gap> tableU:= CharacterTable( "Th" );; gap> fus:= PossibleClassFusions( tableU, table2B );; gap> pi:= Set( fus, > map -> InducedClassFunctionsByFusionMap( tableU, table2B, > [ TrivialCharacter( tableU ) ], map )[1] );; gap> Length( pi ); 1 gap> pi5:= pi[1];; gap> mult:= List( Irr( table2B ), > chi -> ScalarProduct( table2B, chi, pi5 ) );; gap> Maximum( mult ); 2 gap> Positions( mult, 1 ); [ 1, 3, 7, 8, 12, 13, 16, 19, 27, 28, 34, 38, 41, 57, 68, 70, 77, 78, 85, 89, 113, 114, 116, 129, 133, 142, 143, 145, 155, 156, 185, 187, 188, 193, 195, 196, 201, 208, 216, 219, 225, 232, 233, 235, 236, 237, 242 ] gap> Positions( mult, 2 ); [ 62 ] </pre> <div class="p"><!----></div> For U = 2<sup>1+22</sup>.<span class="roman">McL</span>, we carry out the computations described in [<a href="#ctblpope" name="CITEctblpope">Breb</a>,Section "A permutation character of 2 We know that U is a subgroup of 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>, and that 〈U, a 〉 has the structure 2<sup>2+22</sup>.<span class="roman">McL</span>. <div class="p"><!----></div> As a first step, we induce the trivial character of 〈U, a 〉 to 2.<font size="+0">B</font>, which can be performed by inducing the trivial character of <span class="roman">McL</span> to <spa then to inflate this character to 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub>, then to induce this character to <font size="+0">B</font>, and then to inflate this character to 2.<font size="+0">B</font>, <div class="p"><!----></div> <pre> gap> mcl:= CharacterTable( "McL" );; gap> co2:= CharacterTable( "Co2" );; gap> fus:= PossibleClassFusions( mcl, co2 );; gap> Length( fus ); 4 gap> ind:= Set( fus, map -> InducedClassFunctionsByFusionMap( mcl, co2, > [ TrivialCharacter( mcl ) ], map )[1] );; gap> Length( ind ); 1 gap> bm2:= CharacterTable( "BM2" ); CharacterTable( "2^(1+22).Co2" ) gap> infl:= Inflated( co2, bm2, ind );; gap> ind:= Induced( bm2, tableB, infl );; gap> infl:= Inflated( tableB, table2B, ind )[1];; </pre> <div class="p"><!----></div> As a second step, we compute 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> with the <font face="helvetica">GAP</fon using that we can speed up these computations by prescribing the permutation character induced from the closure of U with the normal subgroup 〈a 〉 of 2.<font size="+0">B</font>. <div class="p"><!----></div> (We are lucky: There is a unique solution, and its computation is quite fast.) <div class="p"><!----></div> <pre> gap> centre:= ClassPositionsOfCentre( table2B ); [ 1, 2 ] gap> pi:= PermChars( table2B, rec( torso:= [ 2 * infl[1], 0 ], > normalsubgroup:= centre, > nonfaithful:= infl ) );; gap> Length( pi ); 1 gap> pi6:= pi[1];; gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi6 ) ); [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1, 0, 2, 0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4, 0, 3, 0, 3, 2, 5, 0, 1, 0, 3, 1, 0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, 1, 1, 2, 1, 1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, 0, 0, 2, 0, 0, 0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 3, 2, 0, 0, 2, 2, 4, 3, 5, 2, 4, 0, 0, 0, 0, 5, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 1, 7, 7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, 0, 0, 0, 0, 4, 1, 1, 3, 8, 3, 2, 2, 5, 0, 1 ] </pre> <div class="p"><!----></div> Next we consider U = 2.<span class="roman">F</span><sub>4</sub>(2). We know that U does not contain the central involution of 2.<font size="+0">B</font>. <div class="p"><!----></div> <pre> gap> tableU:= CharacterTable( "2.F4(2)" );; gap> fus:= PossibleClassFusions( tableU, table2B );; gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B, > [ TrivialCharacter( tableU ) ], map )[1] );; gap> Length( pi ); 2 gap> pi:= Filtered( pi, x -> ClassPositionsOfKernel( x ) = [ 1 ] );; gap> Length( pi ); 1 gap> pi7:= pi[1];; gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi7 ) ); [ 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 0, 2, 4, 1, 3, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 1, 4, 0, 2, 1, 2, 0, 0, 3, 2, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 3, 0, 0, 3, 0, 1, 4, 0, 0, 3, 0, 6, 0, 3, 2, 0, 0, 1, 4, 1, 4, 2, 6, 1, 4, 0, 4, 0, 1, 1, 2, 0, 0, 3, 2, 1, 3, 2, 0, 0, 4, 5, 3, 1, 0, 3, 0, 0, 1, 1, 2, 0, 0, 2, 0, 2, 0, 3, 3, 3, 0, 4, 1, 0, 4, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 0, 0, 2, 2, 0, 5, 5, 3, 0, 1, 5, 1, 4, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 2, 3, 1, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 4, 4, 0, 0, 0, 3, 1, 1, 1, 2, 2, 2, 0, 0, 1, 2, 3, 3, 3, 1, 2, 0, 0, 1, 1, 4, 2, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 5, 5, 0, 1, 1, 2, 2, 4, 4, 0, 0, 3, 1, 1, 1, 0, 0, 4, 1, 1, 5, 7, 3, 2, 5, 5, 0, 1 ] </pre> <div class="p"><!----></div> Next we consider U = <span class="roman">HN</span>. <div class="p"><!----></div> <pre> gap> tableU:= CharacterTable( "HN" );; gap> fus:= PossibleClassFusions( tableU, table2B );; gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B, > [ TrivialCharacter( tableU ) ], map )[1] );; gap> Length( pi ); 1 gap> pi8:= pi[1];; gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi8 ) ); [ 1, 1, 2, 1, 2, 0, 3, 4, 2, 1, 1, 4, 4, 2, 1, 1, 3, 3, 1, 3, 0, 0, 5, 3, 0, 0, 6, 4, 5, 6, 1, 7, 4, 7, 0, 0, 3, 8, 2, 6, 11, 2, 5, 5, 0, 0, 2, 1, 3, 4, 7, 0, 0, 7, 3, 9, 5, 0, 0, 6, 4, 2, 13, 6, 0, 4, 4, 12, 11, 16, 9, 7, 3, 11, 13, 12, 20, 5, 10, 6, 11, 13, 17, 4, 10, 7, 19, 7, 7, 8, 10, 14, 18, 19, 5, 10, 12, 23, 7, 12, 6, 24, 6, 4, 17, 16, 8, 9, 17, 11, 12, 23, 8, 24, 18, 26, 21, 29, 10, 18, 31, 10, 24, 21, 17, 27, 35, 13, 14, 29, 19, 12, 7, 18, 26, 15, 34, 34, 35, 20, 14, 36, 14, 39, 8, 29, 24, 15, 40, 13, 9, 38, 24, 17, 35, 32, 26, 26, 24, 22, 17, 31, 39, 29, 30, 30, 19, 44, 37, 37, 28, 30, 31, 29, 42, 40, 40, 56, 56, 30, 30, 42, 50, 47, 2, 2, 4, 6, 4, 0, 0, 4, 6, 10, 10, 12, 8, 12, 0, 0, 2, 4, 16, 10, 0, 0, 2, 12, 10, 0, 0, 0, 0, 0, 0, 28, 0, 0, 14, 34, 40, 2, 10, 10, 22, 40, 44, 44, 8, 8, 36, 14, 14, 16, 8, 8, 46, 28, 28, 58, 90, 72, 70, 92, 104, 56, 90 ] </pre> <div class="p"><!----></div> Finally, we consider U = 2.<span class="roman">Fi</span><sub>22</sub>. There are two candidates for the permutation character (1<sub>U</sub>)<sup>2.<font size="+0">B</fo according to the possible class fusions. One of the two characters is zero on the class of the central involution of 2.<font size="+0">B</font>, the other is not. We know that U does not contain the central involution of 2.<font size="+0">B</font>, hence we can decide which character is correct. <div class="p"><!----></div> <pre> gap> tableU:= CharacterTable( "2.Fi22" );; gap> fus:= PossibleClassFusions( tableU, table2B );; gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B, > [ TrivialCharacter( tableU ) ], map )[1] );; gap> Length( pi ); 2 gap> pi:= Filtered( pi, x -> ClassPositionsOfKernel( x ) = [ 1 ] );; gap> Length( pi ); 1 gap> pi9:= pi[1];; gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi9 ) ); [ 1, 2, 3, 1, 4, 1, 5, 5, 5, 1, 1, 5, 8, 4, 4, 1, 7, 6, 0, 5, 0, 0, 10, 7, 0, 0, 10, 6, 6, 13, 3, 14, 10, 11, 0, 0, 5, 11, 2, 14, 19, 6, 6, 5, 0, 0, 0, 3, 6, 7, 11, 0, 0, 17, 2, 20, 9, 0, 0, 12, 8, 1, 23, 11, 1, 8, 7, 23, 18, 27, 18, 12, 7, 22, 29, 21, 34, 6, 22, 7, 22, 18, 33, 3, 19, 10, 34, 12, 12, 15, 17, 28, 34, 34, 7, 20, 26, 40, 15, 25, 3, 40, 9, 6, 34, 25, 18, 21, 30, 21, 18, 43, 12, 45, 39, 49, 38, 51, 18, 32, 63, 19, 42, 41, 33, 48, 64, 27, 29, 52, 38, 29, 19, 40, 47, 31, 69, 69, 65, 42, 35, 68, 27, 73, 20, 53, 46, 38, 75, 29, 24, 72, 50, 41, 72, 68, 58, 52, 54, 50, 44, 64, 75, 58, 69, 65, 49, 85, 75, 75, 63, 68, 65, 63, 90, 87, 83, 118, 118, 74, 71, 90, 109, 109, 2, 3, 6, 9, 8, 0, 0, 7, 10, 18, 16, 22, 12, 23, 0, 0, 2, 6, 28, 19, 0, 0, 5, 16, 18, 0, 0, 0, 0, 0, 0, 52, 1, 1, 26, 59, 76, 11, 18, 18, 39, 77, 80, 77, 22, 22, 66, 27, 27, 33, 20, 20, 87, 60, 60, 103, 175, 148, 152, 187, 215, 140, 201 ] </pre> <div class="p"><!----></div> Now we can form the restriction of (1<sub>2.<font size="+0">B</font></sub>)<sup><font size="+0">M</fon <div class="p"><!----></div> <pre> gap> constit:= [ pi1, pi2, pi3, pi4, pi5, pi6, pi7, pi8, pi9 ];; gap> pi:= Sum( constit );; </pre> <div class="p"><!----></div> <h2><a name="tth_sEc4"> 4</a> The conjugacy classes of <font size="+0">M</font></h2><a name="Mclasses"> </a> <div class="p"><!----></div> <h3><a name="tth_sEc4.1"> 4.1</a> Our strategy to describe the conjugacy classes of <font size="+0">M</font></h3><a name="stra </a> <div class="p"><!----></div> We know the order of <font size="+0">M</font> and its prime divisors. Let us check whether this fits to our data computed up to now. <div class="p"><!----></div> <pre> gap> sizeM:= pi[1] * Size( table2B ); 808017424794512875886459904961710757005754368000000000 gap> StringPP( sizeM ); "2^46*3^20*5^9*7^6*11^2*13^3*17*19*23*29*31*41*47*59*71" gap> sizeM = Size( CharacterTable( "M" ) ); true </pre> <div class="p"><!----></div> For each prime p dividing |<font size="+0">M</font>|, we classify the conjugacy classes of elements of order p in <font size="+0">M</font> and use the facts that for each such class representative x, the classes of roots of x in the centralizer/normalizer of x are in bijection with the corresponding classes in <font size="+0">M</font>, and that this bijection respects centralizer orders. <div class="p"><!----></div> <div class="p"><!----></div> For each element x ∈ <font size="+0">M</font> of order p ∈ { 2, 3, 5 }, we will use the character table of N<sub><font size="+0">M</font></sub>(〈x 〉) to establish <font size="+0">M</font>-conjugacy classes of roots of x. In order not to count the same class several times, we proceed by increasing p, and collect only those classes of roots of x for which p is the smallest prime divisor of the element order. <div class="p"><!----></div> For elements x ∈ <font size="+0">M</font> of prime order p > 5, it is not necessary to use the character table of N<sub><font size="+0">M</font></sub>(〈x 〉); we will use the permutation character values (1<sub>2.<font size="+0">B</font></sub>)<sup><font size and ad hoc arguments. <div class="p"><!----></div> <h3><a name="tth_sEc4.2"> 4.2</a> Utility functions</h3><a name="functions"> </a> <div class="p"><!----></div> During the process of finding the conjugacy classes of <font size="+0">M</font>, we record our knowledge about the character table of <font size="+0">M</font> in a global <font face="helvetica">GAP</font> variable <tt>head</tt>, which is a record with the following components. <div class="p"><!----></div> <dl compact="compact"> <dt><b><tt>Size</tt></b></dt> <dd> <br /> the group order |<font size="+0">M</font>|,</dd> <dt><b><tt>SizesCentralizers</tt></b></dt> <dd> <br /> the list of centralizer orders of the conjugacy classes established up to now,</dd> <dt><b><tt>OrdersClassRepresentatives</tt></b></dt> <dd> <br /> the list of corresponding representative orders,</dd> <dt><b><tt>fusions</tt></b></dt> <dd> <br /> a list that collects the currently known partial class fusions into <font size="+0">M</font>; each entry is a record with the components <tt>subtable</tt> (the character table of the subgroup) and <tt>map</tt> (the list of known images; unknown positions are unbound).</dd> </dl> <div class="p"><!----></div> We initialize this variable, using the group order <font size="+0">M</font> and that there is an identity element. <div class="p"><!----></div> <pre> gap> head:= rec( Size:= sizeM, > SizesCentralizers:= [ sizeM ], > OrdersClassRepresentatives:= [ 1 ], > fusions:= [], > );; </pre> <div class="p"><!----></div> The function <tt>ExtendTableHeadByRootClasses</tt> takes the object <tt>head</tt>, the character table <tt>s</tt> of a subgroup H of <font size="+0">M</font>, and an integer <tt>pos</tt> as its arguments, where it is assumed that the <tt>pos</tt>-th class of <tt>s</tt> contains an element x of prime order p such that N<sub><font size="+0">M</font></sub>(〈x 〉) = H holds and such that <tt>head</tt> contains information only about those classes of <font size="+0">M</font> whose elements have order divisible by a prime that is smaller than p. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByRootClasses:= function( head, s, pos ) > local fus, orders, p, cents, oldnumber, i, ord; > > # Initialize the fusion information. > fus:= rec( subtable:= s, map:= [ 1 ] ); > Add( head.fusions, fus ); > > # Compute the positions of root classes of 'pos'. > orders:= OrdersClassRepresentatives( s ); > p:= orders[ pos ]; > cents:= SizesCentralizers( s ); > oldnumber:= Length( head.OrdersClassRepresentatives ); > > # Run over the classes of 's' > # are already contained in head > for i in [ 1 .. NrConjugacyClasses( s ) ] do > ord:= orders[i]; > if ord mod p = 0 and > Minimum( PrimeDivisors( ord ) ) = p and > PowerMap( s, ord / p, i ) = pos then > # Class 'i' is a root class of 'pos' and is new in 'head'. > Add( head.SizesCentralizers, cents[i] ); > Add( head.OrdersClassRepresentatives, orders[i] ); > fus.map[i]:= Length( head.SizesCentralizers ); > fi; > od; > > Print( "#I after ", Identifier( s ), ": found ", > Length( head.OrdersClassRepresentatives ) - oldnumber, > " classes, now have ", > Length( head.OrdersClassRepresentatives ), "\n" ); > end;; </pre> <div class="p"><!----></div> In several cases, we will establish a conjugacy class g<sup><font size="+0">M</font></sup> without knowing the character table of a suitable subgroup of <font size="+0">M</font> to which <tt>ExtendTableHeadByRootClasses</tt> can be applied, where g is among the root classes. That is, we may know just element order <tt>s</tt> and centralizer order <tt>cent</tt>. <div class="p"><!----></div> We are a bit better off if we know the character table <tt>s</tt> of a subgroup of <font size="+0">M</font> and the list <tt>poss</tt> of all those classes in this table which fuse to the class g<sup><font size="+0">M</font></sup>, because then we can store this information in the partial class fusion from <tt>s</tt> that is stored in <tt>head</tt>. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByCentralizerOrder:= function( head, s, cent, poss ) > local ord, fus, i; > > if IsCharacterTable( s ) then > ord:= Set( OrdersClassRepresentatives( s ){ poss } ); > if Length( ord ) <> 1 then > Error( "classes cannot fuse" ); > fi; > ord:= ord[1]; > elif IsInt( s ) then > ord:= s; > fi; > Add( head.SizesCentralizers, cent ); > Add( head.OrdersClassRepresentatives, ord ); > > Print( "#I after order ", ord, " element" ); > if IsCharacterTable( s ) then > # extend the stored fusion from s > fus:= First( head.fusions, > r -> Identifier( r.subtable ) = Identifier( s ) ); > for i in poss do > fus.map[i]:= Length( head.SizesCentralizers ); > od; > Print( " from ", Identifier( s ) ); > fi; > Print( ": have ", > Length( head.OrdersClassRepresentatives ), " classes\n" ); > end;; </pre> <div class="p"><!----></div> The permutation character 1<sub>H</sub><sup>G</sup>, where H ≤ G are two groups, has the property 1<sub>H</sub><sup>G</sup>(g) = |C<sub>G</sub>(g)| ·|g<sup>G</sup> ∩H| / |H|. For g ∈ H, this implies that |C<sub>G</sub>(g)| = 1<sub>H</sub><sup>G</sup>(g) ·|H| / |g<sup>G</sup> ∩H| can be computed from the character (1<sub>H</sub><sup>G</sup>)<sub>H</sub> and the class lengths in H, provided that we know which classes of H fuse into g<sup>G</sup>. The function <tt>ExtendTableHeadByPermCharValue</tt> extends the information in <tt>head</tt> by the data for the class g<sup><font size="+0">M</font></sup>, where <tt>s</tt> is the character table of H, <tt>pi_rest_to_s</tt> is (1<sub>H</sub><sup>G</sup>)<sub>H</sub>, and <tt>poss</tt> is the list of positions of those classes in <tt>s</tt> that fuse to g<sup><font size="+0">M</font></sup>. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByPermCharValue:= function( head, s, pi_rest_to_s, poss ) > local pival, cent; > > pival:= Set( pi_rest_to_s{ poss } ); > if Length( pival ) <> 1 then > Error( "classes cannot fuse" ); > fi; > > cent:= pival[1] * Size( s ) / Sum( SizesConjugacyClasses( s ){ poss } ); > ExtendTableHeadByCentralizerOrder( head, s, cent, poss ); > end;; </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.3"> 4.3</a> Classes of elements of even order</h3><a name="elements_2"> </a> <div class="p"><!----></div> By [<a href="#Mverify" name="CITEMverify">BMW24</a>], we know that <font size="+0">M</font> has exactly two conjugacy classes of involutions, and that the involution centralizers have the structures 2.<font size="+0">B</font> (for the class <tt>2A</tt>) and 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub> (for the class <tt>2B</tt>), respec <div class="p"><!----></div> Moreover, the character tables of these subgroups that are stored in the <font face="helvetica">GAP</font> Character Table Library are correct. For 2.<font size="+0">B</font>, this follows from the correctness of the character table of <font s as shown in [<a href="#BMverify" name="CITEBMverify">BMW20</a>] and the computations in []. For 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>, the recomputation of the ch in Section <a href="#sect:table_c2b">8</a>. <div class="p"><!----></div> Thus we can determine the <font size="+0">M</font>-conjugacy classes of elements of even order as follows. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2.B" );; gap> ClassPositionsOfCentre( s ); [ 1, 2 ] gap> ExtendTableHeadByRootClasses( head, s, 2 ); #I after 2.B: found 42 classes, now have 43 gap> s:= CharacterTable( "MN2B" );; gap> ClassPositionsOfCentre( s ); [ 1, 2 ] gap> ExtendTableHeadByRootClasses( head, s, 2 ); #I after 2^1+24.Co1: found 91 classes, now have 134 </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.4"> 4.4</a> Classes of elements of order divisible by 3</h3><a name="elements_3"> </a> <div class="p"><!----></div> We know that <font size="+0">M</font> has exactly three conjugacy classes of elements of order 3, and that their normalizers have the structures 3.<span class="roman">Fi</span><sub>24</sub> (for the class <tt>3A</tt>), 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 (for the class <tt>3B</tt>), and S<sub>3</sub> ×<span class="roman">Th</span> (for the class <tt>3C</tt>), respectively. <div class="p"><!----></div> Moreover, the <font face="helvetica">GAP</font> character tables of 3.<span class="roman">Fi</span><sub>24</ and have been verified, see [<a href="#BMO17" name="CITEBMO17">BMO17</a>]. <div class="p"><!----></div> We determine the <font size="+0">M</font>-conjugacy classes of elements of odd order that are roots of <tt>3A</tt> or <tt>3C</tt> elements, as follows. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "3.Fi24" );; gap> ClassPositionsOfPCore( s, 3 ); [ 1, 2 ] gap> ExtendTableHeadByRootClasses( head, s, 2 ); #I after 3.F3+.2: found 12 classes, now have 146 gap> s:= CharacterTableDirectProduct( CharacterTable( "Th" ), > CharacterTable( "Symmetric", 3 ) );; gap> ClassPositionsOfPCore( s, 3 ); [ 1, 3 ] gap> ExtendTableHeadByRootClasses( head, s, 3 ); #I after ThxSym(3): found 7 classes, now have 153 </pre> <div class="p"><!----></div> The situation with the <tt>3B</tt> normalizer is more involved. Section <a href="#sect:norm3B">9</a> documents the construction of the character table of a downward extension of the structure 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</sp of the <tt>3B</tt> normalizer, and gives two candidates for the character table of the <tt>3B</tt> normalizer. <div class="p"><!----></div> It will turn out that each of these candidates leads to "the same" root classes, in the sense that the number of these classes, their element orders, and their centralizer orders are equal. Note that the 3-core of H = 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2 has the str X ×N, where X has order 3 and N ≅ 3<sup>1+12</sup><sub>+</sub> such that H / N ≅ 6.<span class="roman">Suz</span>.2 holds. We are interested in the two "diagonal" factors, that is, the factors of H by the one of the two normal subgroups of order 3 in H that are not equal to X or Z(N). (See the picture in Section <a href="#sect:norm3B">9</a> for the details.) <div class="p"><!----></div> First we exclude the normal subgroup of order 3 that is contained in the unique normal subgroup N of order 3<sup>13</sup>. <div class="p"><!----></div> <pre> gap> exts:= CharacterTable( "3^(1+12):6.Suz.2" );; gap> kernels:= Positions( SizesConjugacyClasses( exts ), 2 ); [ 2, 18, 19, 20 ] gap> order3_13:= Filtered( ClassPositionsOfNormalSubgroups( exts ), > l -> Sum( SizesConjugacyClasses( exts ){ l } ) = 3^13 ); [ [ 1 .. 4 ] ] gap> kernels:= Difference( kernels, order3_13[1] ); [ 18, 19, 20 ] </pre> <div class="p"><!----></div> The classes in the subgroup X can be identified by the fact that exactly one factor of H by a normal subgroup of order 3 admits a class fusion from 2.<span class="roman">Suz</span>.2, and hence this must be the split extension of 3<sup>1+12</sup><sub>+</sub> with 2.<span class="roman">Suz</span>.2. <div class="p"><!----></div> <div class="p"><!----></div> <pre> gap> facts:= List( kernels, i -> exts / [ 1, i ] ); [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 18 ]" ), CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ), CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" ) ] gap> f:= CharacterTable( "2.Suz.2" );; gap> facts:= Filtered( facts, > x -> Length( PossibleClassFusions( f, x ) ) = 0 ); [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ), CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" ) ] </pre> <div class="p"><!----></div> We compute the root classes for both candidates. For that, we first create a copy <tt>head2</tt> of the information in <tt>head</tt>. <div class="p"><!----></div> <pre> gap> kernels:= List( facts, > f -> Positions( SizesConjugacyClasses( f ), 2 ) ); [ [ 2 ], [ 2 ] ] gap> head2:= StructuralCopy( head );; gap> ExtendTableHeadByRootClasses( head, facts[1], 2 ); #I after 3^(1+12):6.Suz.2/[ 1, 19 ]: found 12 classes, now have 165 gap> ExtendTableHeadByRootClasses( head2, facts[2], 2 ); #I after 3^(1+12):6.Suz.2/[ 1, 20 ]: found 12 classes, now have 165 </pre> <div class="p"><!----></div> We observe that <tt>head</tt> and <tt>head2</tt> differ only by the two character tables in the last fusion record. <div class="p"><!----></div> <pre> gap> nams:= RecNames( head ); [ "Size", "OrdersClassRepresentatives", "SizesCentralizers", "fusions" ] gap> ForAll( Difference( nams, [ "fusions" ] ), > nam -> head.( nam ) = head2.( nam ) ); true gap> Length( head.fusions ); 5 gap> ForAll( [ 1 .. 4 ], i -> head.fusions[i] = head2.fusions[i] ); true gap> head.fusions[5].map = head2.fusions[5].map; true </pre> <div class="p"><!----></div> We continue with establishing the conjugacy classes of <font size="+0">M</font>. The question which of the two above candidate tables belongs to a subgroup of <font size="+0">M</font> will be answered in Section <a href="#sect:natcharM">6</a>. <div class="p"><!----></div> <h3><a name="tth_sEc4.5"> 4.5</a> Classes of elements of order divisible by 5</h3><a name="elements_5"> </a> <div class="p"><!----></div> The group 2.<font size="+0">B</font> contains two rational conjugacy classes of elements of order 5, with different values in the permutation character (1<sub>2.<font size="+0">B</font></sub>)<sup><fo <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2.B" );; gap> pos:= Positions( OrdersClassRepresentatives( s ), 5 ); [ 23, 25 ] gap> pi{ pos }; [ 1539000, 7875 ] </pre> <div class="p"><!----></div> This establishes two classes <tt>5A</tt>, <tt>5B</tt> of conjugacy classes of elements of order 5 in <font size="+0">M</font>, with centralizer orders 5 |<span class="roman">HN</span>| and 5<sup>7</sup> |2.<span class="roman">J</s <div class="p"><!----></div> <pre> gap> cents:= List( pos, > i -> pi[i] * Size( s ) / SizesConjugacyClasses( s )[i] ); [ 1365154560000000, 94500000000 ] gap> cents = [ 5 * Size( CharacterTable( "HN" ) ), > 5^7 * Size( CharacterTable( "2.J2" ) ) ]; true </pre> <div class="p"><!----></div> <div class="p"><!----></div> By [<a href="#Mverify" name="CITEMverify">BMW24</a>], we know that <font size="+0">M</font> contains exactly two conjugacy classes of elements of order 5, <tt>5A</tt> with centralizer 5 ×<span class="roman">HN</span> and normalizer (D<sub>10</sub> ×<span class="roman">HN</span>).2, and <tt>5B</tt> with centralizer 5<sup>1+6</sup><sub>+</sub>.2.<span class="roman">J</span><sub>2</sub> an 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2. <div class="p"><!----></div> The two classes are rational because this is the case already for their intersections with 2.<font size="+0">B</font>. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "MN5A" ); CharacterTable( "(D10xHN).2" ) gap> ClassPositionsOfPCore( s, 5 ); [ 1, 45 ] gap> ExtendTableHeadByRootClasses( head, s, 45 ); #I after (D10xHN).2: found 5 classes, now have 170 </pre> <div class="p"><!----></div> The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2 has bee with <font face="helvetica">MAGMA</font>, see Section <a href="#sect:table_N5B">10</a>, thus we are allowed to use the character table from the <font face="helvetica">GAP</font> character table library. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "MN5B" ); CharacterTable( "5^(1+6):2.J2.4" ) gap> 5core:= ClassPositionsOfPCore( s, 5 ); [ 1 .. 4 ] gap> SizesConjugacyClasses( s ){ 5core }; [ 1, 4, 37800, 40320 ] gap> ExtendTableHeadByRootClasses( head, s, 2 ); #I after 5^(1+6):2.J2.4: found 3 classes, now have 173 </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.6"> 4.6</a> Classes of elements of order divisible by 11</h3><a name="elements_11"> </a> <div class="p"><!----></div> The group 2.<font size="+0">B</font> contains a rational class of elements of order 11. The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup>)< of order 11 with centralizer order 11 |M<sub>12</sub>| in <font size="+0">M</font>. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2.B" );; gap> pos:= Positions( OrdersClassRepresentatives( s ), 11 ); [ 71 ] </pre> <div class="p"><!----></div> By the arguments in [<a href="#Mverify" name="CITEMverify">BMW24</a>], <font size="+0">M</font> has no other classes of element order 11. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByPermCharValue( head, s, pi, pos ); #I after order 11 element from 2.B: have 174 classes </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.7"> 4.7</a> Classes of elements of the orders 17, 19, 23, 31, 47</h3><a name="elements_17"> </a> <div class="p"><!----></div> The elements of the orders 17, 19, 23, 31, 47 in <font size="+0">M</font> lie in cyclic Sylow subgroups that appear already in 2.<font size="+0">B</font>. <div class="p"><!----></div> The elements of order 17 and 19 are rational in 2.<font size="+0">B</font> and hence also in <font size="+0">M</font>. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2.B" );; gap> pos:= Positions( OrdersClassRepresentatives( s ), 17 ); [ 118 ] gap> ExtendTableHeadByPermCharValue( head, s, pi, pos ); #I after order 17 element from 2.B: have 175 classes gap> pos:= Positions( OrdersClassRepresentatives( s ), 19 ); [ 128 ] gap> ExtendTableHeadByPermCharValue( head, s, pi, pos ); #I after order 19 element from 2.B: have 176 classes </pre> <div class="p"><!----></div> For elements g of order p ∈ { 23, 31, 47 }, the group 2.<font size="+0">B</font> contains exactly two Galois conjugate classes that contain the nonidentity powers of g, which means that [N<sub>2.<font size="+0">B</font></sub>(〈g 〉):C<sub>2.<font size="+0">B</font></sub>(g) The equation |C<sub><font size="+0">M</font></sub>(g)| = |2.<font size="+0">B</font>| ·π(g) / |g<sup><font size= implies <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="cente |N<sub><font size="+0">M</font></sub>(〈g 〉)| = [N<sub><font size="+0">M</font></sub>(〈g 〉):C<sub><font size="+0"> </td></tr></table> Note that either the two classes of elements of order p in 2.<font size="+0">B</font> fuse in <font size="+0">M</font> or not; in the former case, we have [N<sub><font size="+0">M</font></sub>(〈g 〉):C<sub><font size="+0">M</font></sub>(g)] = p−1 and |g<sup><font size="+0">M</font></sup> ∩2.<font size="+0">B</font>| = 2 |g<sup>2.<font size="+0">B</font></sup>|, whereas we have [N<sub><font size="+0">M</font></sub>(〈g 〉):C<sub><font size="+0">M</font></sub>(g)] = (p−1)/2 and |g<sup><font size="+0">M</font></sup> ∩2.<font size="+0">B</font>| = |g<sup>2.<font size="+0">B</font></sup>| in Thus we can compute |N<sub><font size="+0">M</font></sub>(〈g 〉)| in each case, and we can then find arguments why the two Galois conjugate classes do not fuse. <div class="p"><!----></div> First we deal with p = 23. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2.B" ); CharacterTable( "2.B" ) gap> ord:= OrdersClassRepresentatives( s );; gap> classes:= SizesConjugacyClasses( s );; gap> p:= 23;; gap> pos:= Positions( ord, p ); [ 147, 149 ] gap> n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ]; 6072 gap> Collected( Factors( n ) ); [ [ 2, 3 ], [ 3, 1 ], [ 11, 1 ], [ 23, 1 ] ] </pre> <div class="p"><!----></div> In order to prove that the two classes of elements of order 23 in 2.<font size="+0">B</font> do not fuse in <font size="+0">M</font>, it suffices to show that the centralizer order is divisible by 2<sup>3</sup>. We see that this is the case already in the <tt>2B</tt> centralizer in <font size="+0">M</font>. <div class="p"><!----></div> <pre> gap> u:= CharacterTable( "MN2B" ); CharacterTable( "2^1+24.Co1" ) gap> upos:= Positions( OrdersClassRepresentatives( u ), p ); [ 289, 294 ] gap> SizesCentralizers( u ){ upos } / 2^3; [ 23, 23 ] </pre> <div class="p"><!----></div> Thus we have established two classes of element order 23 in <font size="+0">M</font>. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } ); #I after order 23 element from 2.B: have 177 classes gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } ); #I after order 23 element from 2.B: have 178 classes </pre> <div class="p"><!----></div> The case p = 31 is done analogously. Here the necessary 2-part of the centralizer occurs already in 2.<font size="+0">B</font>. <div class="p"><!----></div> <pre> gap> p:= 31;; gap> pos:= Positions( OrdersClassRepresentatives( s ), p ); [ 190, 192 ] gap> n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ]; 2790 gap> Collected( Factors( n ) ); [ [ 2, 1 ], [ 3, 2 ], [ 5, 1 ], [ 31, 1 ] ] gap> SizesCentralizers( s ){ pos }; [ 62, 62 ] gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } ); #I after order 31 element from 2.B: have 179 classes gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } ); #I after order 31 element from 2.B: have 180 classes </pre> <div class="p"><!----></div> Finally, we deal with p = 47. <div class="p"><!----></div> <pre> gap> p:= 47;; gap> pos:= Positions( OrdersClassRepresentatives( s ), p ); [ 228, 230 ] gap> n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ]; 2162 gap> Collected( Factors( n ) ); [ [ 2, 1 ], [ 23, 1 ], [ 47, 1 ] ] gap> SizesCentralizers( s ){ pos }; [ 94, 94 ] gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } ); #I after order 47 element from 2.B: have 181 classes gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } ); #I after order 47 element from 2.B: have 182 classes </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.8"> 4.8</a> Classes of elements of order 13</h3><a name="elements_13"> </a> <div class="p"><!----></div> The class <tt>13A</tt> of <font size="+0">M</font> arises from the rational class of elements of order 13 in 2.<font size="+0">B</font>. We use the permutation character to enter the information about the class <tt>13A</tt>. <div class="p"><!----></div> <pre> gap> p:= 13;; gap> pos:= Positions( OrdersClassRepresentatives( s ), p ); [ 97 ] gap> c:= Size( s ) * pi[ pos[1] ] / classes[ pos[1] ]; 73008 gap> Factors( c ); [ 2, 2, 2, 2, 3, 3, 3, 13, 13 ] gap> ExtendTableHeadByPermCharValue( head, s, pi, pos ); #I after order 13 element from 2.B: have 183 classes </pre> <div class="p"><!----></div> The class <tt>13B</tt> intersects the <tt>2B</tt> centralizer. Here we just know the centralizer order 13<sup>3</sup> ·2<sup>3</sup> ·3. <div class="p"><!----></div> <pre> gap> c2b:= CharacterTable( "MN2B" );; gap> pos:= Positions( OrdersClassRepresentatives( c2b ), 13 ); [ 220 ] gap> ExtendTableHeadByCentralizerOrder( head, c2b, 13^3 * 24, pos ); #I after order 13 element from 2^1+24.Co1: have 184 classes </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.9"> 4.9</a> Classes of elements of order divisible by 29</h3><a name="elements_29"> </a> <div class="p"><!----></div> The group 3.<span class="roman">Fi</span><sub>24</sub> contains a rational class of elements of ord with centralizer order 3 ·29. <div class="p"><!----></div> <pre> gap> u:= CharacterTable( "3.Fi24" );; gap> pos:= Positions( OrdersClassRepresentatives( u ), 29 ); [ 142 ] gap> SizesCentralizers( u ){ pos }; [ 87 ] </pre> <div class="p"><!----></div> The list of classes of <font size="+0">M</font> collected up to now covers all roots of elements of the orders 2, 3, 5, 11, 13, 17, 19, 23, 31, 47, and 29 occurs as a factor of the centralizer order only for the classes <tt>1A</tt>, <tt>3A</tt>, <tt>87A</tt>, and <tt>87B</tt>. <div class="p"><!----></div> <pre> gap> poss:= PositionsProperty( head.SizesCentralizers, > x -> x mod 29 = 0 ); [ 1, 135, 144, 145 ] gap> head.OrdersClassRepresentatives{ poss }; [ 1, 3, 87, 87 ] gap> head.SizesCentralizers{ poss }; [ 808017424794512875886459904961710757005754368000000000, 3765617127571985163878400, 87, 87 ] </pre> <div class="p"><!----></div> Thus the only possible additional prime divisors of the centralizer order in <font size="+0">M</font> of an element x of order 29 are 7, 41, 59, and 71. <div class="p"><!----></div> <pre> gap> candprimes:= Difference( PrimeDivisors( head.Size ), > [ 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 47 ] ); [ 7, 41, 59, 71 ] </pre> <div class="p"><!----></div> The centralizer order of x has the form 3 ·29 ·7<sup>i</sup> ·41<sup>j</sup> ·59<sup>k</sup> ·71<sup>l</sup>, with 0 ≤ i ≤ 6 and j, k, l ∈ { 0, 1 }. <div class="p"><!----></div> <pre> gap> parts:= Filtered( Collected( Factors( head.Size ) ), > x -> x[1] in candprimes ); [ [ 7, 6 ], [ 41, 1 ], [ 59, 1 ], [ 71, 1 ] ] gap> poss:= List( parts, l -> List( [ 0 .. l[2] ], i -> l[1]^i ) );; gap> cart:= Cartesian( poss );; gap> possord:= 3 * 29 * List( cart, Product );; </pre> <div class="p"><!----></div> Only 3 ·29 and 3 ·29 ·59 satisfy Sylow's theorem, that is, |<font size="+0">M</font>| / |N<sub><font size="+0">M</font></sub>(〈x 〉)| ≡ 1 mod 29. Note that we have [N<sub><font size="+0">M</font></sub>(〈x 〉):C<sub><font size="+0">M</font></sub>(x)] = 28 <div class="p"><!----></div> <pre> gap> good:= Filtered( possord, > x -> ( head.Size / ( 28 * x ) ) mod 29 = 1 ); [ 87, 5133 ] gap> List( good, Factors ); [ [ 3, 29 ], [ 3, 29, 59 ] ] </pre> <div class="p"><!----></div> Now we can exclude the possible centralizer order 3 ·29 ·59 by the fact that the Sylow 59 subgroup would be normal and thus would be normalized and hence centralized by an element of order 3, a contradiction. <div class="p"><!----></div> <pre> gap> Filtered( DivisorsInt( 5133 ), x -> x mod 59 = 1 ); [ 1 ] </pre> <div class="p"><!----></div> Thus we have established a rational class of elements of order 29, with centralizer of order 3 ·29. <div class="p"><!----></div> <pre> gap> ExtendTableHeadByCentralizerOrder( head, u, 3 * 29, pos ); #I after order 29 element from 3.F3+.2: have 185 classes </pre> <div class="p"><!----></div> <h3><a name="tth_sEc4.10"> 4.10</a> Classes of elements of order divisible by 41</h3><a name="elements_41"> </a> <div class="p"><!----></div> --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.42 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28