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< title>Some steps in the verification of the ordinary character table of the Monster group</ title>
< h1 align= "center">Some steps in the verification of the ordinary character table of the Monste r group</h1>
<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">AY</font> M<font size="-2">AGAARD</font>, R<font size="-2">OBERT</font> A. W<font size="-2">ILSON</font> </h3>
<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="CITEMverify">BMW24</a>].
<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>)<sub>2.<font size="+0">B</font></sub></a><br /><a href="#tth_sEc4"
>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_sEc4.2"
>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 967; 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</sub></a><br /><a href="#tth_sEc9"
>9 Appendix: The character table of 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2</a><br /> <a href="#tth_sEc9.1"
>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</sub>.2</a><br />
<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 character table
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> style='color: green'>8242;</sup> (and 3.<span class="roman">Fi</span><sub>24</sub>),
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) 8773; 2.<font size="+0">B</font> on the conjugacy class a<sup><font size="+0">M</font></sup>,
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</span><sub>1</sub>, 3.<span class="roman">Fi</span><sub>24</sub>, and
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 three
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" name="CITEAGRv3">WWT<sup>+</sup></a>],
thus we load these <font face="helvetica">GAP</font> packages.
<div class="p"><!----></div>
<pre>
gap62; LoadPackage( "ctbllib", false );
true
gap62; LoadPackage( "atlasrep", false );
true
</pre>
<div class="p"><!----></div>
The <font face="helvetica">MAGMA</font> system [<a href="#Magma" name="CITEMagma">BCP97</a>] will be needed
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>
gap62; 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>
gap62; 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 967;
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 967; 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>,Lemma 1].
This follows from the following data about 2.<font size="+0">B</font>.
<div class="p"><!----></div>
<pre>
gap62; table2B:= CharacterTable( "2.B" );;
gap62; cand:= Filtered( Irr( table2B ), x -62; x[1] <= 196883 );;
gap62; List( cand, x -62; x[1] );
[ 1, 4371, 96255, 96256 ]
gap62; inv:= Positions( OrdersClassRepresentatives( table2B ), 2 );
[ 2, 3, 4, 5, 7 ]
gap62; PrintArray( List( cand, x ->62; 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 + 2 183;96256a
or 4371 183;1a + 2 183;96256a would be possible decompositions,
which have more than two values on involution classes.
Thus the values of 967;<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 967;<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 183;1a + 8 183;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 967; on the classes of involutions are
4371 and 275, respectively.
<div class="p"><!----></div>
<pre>
gap62; Sum( cand ){ inv };
[ 4371, 4371, 4371, 275, 275 ]
</pre>
<div class="p"><!----></div>
The restriction of 967; to 3.<span class="roman">Fi</span><sub>24</sub><sup>8242;</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>yle='color: green'>8242;</sup> can occur as
constituents of the restriction of 967;.
<div class="p"><!----></div>
<pre>
gap62; table3Fi24prime:= CharacterTable( "3.Fi24'" );;
gap62; cand:= Filtered( Irr( table3Fi24prime ), x -='color: green'>62; x[1] <= 196883 );;
gap62; inv:= Positions( OrdersClassRepresentatives( table3Fi24prime ), 2 );
[ 4, 7 ]
gap62; mat:= List( cand, x -62; x{ Concatenation([1], inv)});;
gap62; 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 967; 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>
gap62; List( cand, x -62; 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 2 183;64584 exist.
<div class="p"><!----></div>
<pre>
gap62; Float( 196883 / 4371 );
45.043
gap62; List( mat, v -62; Float( v[1] / v[2] ) );
[ 1., 24.7037, 49.6776, 9.91139, 9.91139, 47.7692, 47.7692 ]
gap62; Float( ( 196883 - 2 * 64584 ) / ( 4371 - 2 * 1352 ) );
40.6209
gap62; Float( ( 196883 - 57477 ) / ( 4371 - 1157 ) );
43.3746
gap62; Float( ( 196883 - 2*57477 ) / ( 4371 - 2*1157 ) );
39.8294
gap62; 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>
gap62; 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>
gap62; 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>8242;</sup> to 3.<span class="roman">Fi</span><sub>24</sub>.
In order to decompose the restriction of 967; 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>tyle='color: green'>8242;</sup> occur.
The following irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub> can occur as
constituents of the restriction of 967;.
<div class="p"><!----></div>
<pre>
gap62; table3Fi24:= CharacterTable( "3.Fi24" );;
gap62; cand:= Filtered( Irr( table3Fi24 ), x -62; x[1] <= 196883 );;
gap62; inv:= Positions( OrdersClassRepresentatives( table3Fi24 ), 2 );
[ 3, 5, 172, 173 ]
gap62; mat:= List( cand, x -62; x{ Concatenation([1], inv)});;
gap62; 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>,Lemma 2].
<div class="p"><!----></div>
<pre>
gap62; 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></sup>)<sub>2.<font size="+0">B</font></sub></h2><a name="suborbits">
</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="+0">M</font></sup> to 2.<font size="+0">B</font>
decomposes into nine transitive permutation characters 1<sub>U</sub><sup>2.<font size="+0">B</font></sup>,
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 8712; </td><td>G<sub>a,c</sub> </td><td align="right">e='color: green'>124;c<sup>G<sub>a</sub></sup>124; </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="right">27143910000 </td></tr>
<tr><td><tt>2B</tt> </td><td>2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> </td><td align="right">11707448673375 </td></tr>
<tr><td><tt>3A</tt> </td><td><span class="roman">Fi</span><sub>23</sub> </td><td align="right">2031941058560000 </td></tr>
<tr><td><tt>3C</tt> </td><td><span class="roman">Th</span> </td><td align="right">91569524834304000 </td></tr>
<tr><td><tt>4A</tt> </td><td>2<sup>1+22</sup>.<span class="roman">McL</span> </td><td align="right">1102935324621312000 </td></tr>
<tr><td><tt>4B</tt> </td><td>2.<span class="roman">F</span><sub>4</sub>(2) </td><td align="right">1254793905192960000 </td></tr>
<tr><td><tt>5A</tt> </td><td><span class="roman">HN</span> </td><td align="right">30434513446055706624 </td></tr>
<tr><td><tt>6A</tt> </td><td>2.<span class="roman">Fi</span><sub>22</sub> </td><td align="right">64353605265653760000 </td></tr></table>
</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 8712; 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 8712; 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>.<span class="roman">McL</span>
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 subgroup
2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub> of <font size="+0">B</font> under the natural epimorphism from 2.<font size="+0">B</font>,
and the computation/verification of the character table of 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub>
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</span>.
<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>
gap62; 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>[='color: green'>63717;U]</sub><sup><font size="+0">B</font></sup>
from <font size="+0">B</font> to 2.<font size="+0">B</font>,
for [63717;U] = U / 9001;a yle='color: green'>9002; = 2.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2).
(Note that the class fusion from [63717;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>
gap62; tableB:= CharacterTable( "B" );;
gap62; tableUbar:= CharacterTable( "2.2E6(2)" );;
gap62; fus:= PossibleClassFusions( tableUbar, tableB );;
gap62; pi:= Set( fus,
62; map -62; InducedClassFunctionsByFusionMap( tableUbar, tableB,
62; [ TrivialCharacter( tableUbar ) ], map )[1] );;
gap62; Length( pi );
1
gap62; pi2:= Inflated( tableB, table2B, pi )[1];;
gap62; mult:= List( Irr( table2B ),
62; chi -62; ScalarProduct( table2B, chi, pi2 ) );;
gap62; Maximum( mult );
1
gap62; 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>[='color: green'>63717;U]</sub><sup><font size="+0">B</font></sup>
from <font size="+0">B</font> to 2.<font size="+0">B</font>,
for [63717;U] = U / 9001;a yle='color: green'>9002; = 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>
gap62; tableUbar:= CharacterTable( "BN2B" );
CharacterTable( "2^(1+22).Co2" )
gap62; fus:= PossibleClassFusions( tableUbar, tableB );;
gap62; pi:= Set( fus,
62; map -62; InducedClassFunctionsByFusionMap( tableUbar, tableB,
62; [ TrivialCharacter( tableUbar ) ], map )[1] );;
gap62; Length( pi );
1
gap62; pi3:= Inflated( tableB, table2B, pi )[1];;
gap62; mult:= List( Irr( table2B ),
62; chi -62; ScalarProduct( table2B, chi, pi3 ) );;
gap62; Maximum( mult );
1
gap62; 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>
gap62; tableU:= CharacterTable( "Fi23" );;
gap62; fus:= PossibleClassFusions( tableU, table2B );;
gap62; pi:= Set( fus,
62; map -62; InducedClassFunctionsByFusionMap( tableU, table2B,
62; [ TrivialCharacter( tableU ) ], map )[1] );;
gap62; Length( pi );
1
gap62; pi4:= pi[1];;
gap62; mult:= List( Irr( table2B ),
62; chi -62; ScalarProduct( table2B, chi, pi4 ) );;
gap62; Maximum( mult );
1
gap62; 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>
gap62; tableU:= CharacterTable( "Th" );;
gap62; fus:= PossibleClassFusions( tableU, table2B );;
gap62; pi:= Set( fus,
62; map -62; InducedClassFunctionsByFusionMap( tableU, table2B,
62; [ TrivialCharacter( tableU ) ], map )[1] );;
gap62; Length( pi );
1
gap62; pi5:= pi[1];;
gap62; mult:= List( Irr( table2B ),
62; chi -62; ScalarProduct( table2B, chi, pi5 ) );;
gap62; Maximum( mult );
2
gap62; 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 ]
gap62; 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.<font size="+0">B</font>"].
We know that U is a subgroup of 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>,
ndjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 95
<div class="p"><!----></div>
As a first step, we induce the trivial character of 9001;U, a style='color: green'>9002;
to 2.<font size="+0">B</font>,
which can be performed by inducing the trivial character of <span class="roman">McL</span> to <span class="roman">Co</span><sub>2</tddiv.comb{: -. - -ex}
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 this character to .font "0"B/font>,
<div class="p"><!----></div>
<pre>
gap62; mcl:= CharacterTable( "McL" );;
gap62; co2:= CharacterTable( "Co2" );;
gap62; fus:= PossibleClassFusions( mcl, co2 );;
gap62; Length( fus );
4
gap62; ind:= Set( fus, map -62; InducedClassFunctionsByFusionMap( mcl, co2,
62; [ TrivialCharacter( mcl ) ], map )[1] );;
gap62; Length( ind );
1
gap62; bm2:= CharacterTable( "BM2" );
CharacterTable("^(+22)Co2" )
gap62; infl:= Inflated( co2, bm2, ind );;
gap62; ind:= Induced( bm2, tableB, infl );;
gap62; infl:= Inflated( tableB, table2B, ind )[1];;
</pre>
As a second step,
we compute 1<sub>U</sub><sup>2.<font size="+0">B</span.overacc1 {position/style
java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 60
thecharacter from closure U
the #001; &9002;of2.font=+"B/font>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(We are
There java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="p"><!----></div>
<pre>
gap62; centre:= ClassPositionsOfCentre( table2B );
[ 1, 2 ]
.2 /< /&;n;&;href"java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 83
&62 =,
62; nonfaithful:= infl ) );;
gap#;Length(pi )
>6n; of of orderdivisible 11<>< >&;n;;n;< href"tth_sEc4.7java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
gap62; pi6:= pi[1];;
gap62; List( Irr( table2B ), chi -62; ScalarProduct( >8n;Classeselements 13/a />;nbsp &;a="#9
[ 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>.&java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 84
, 0 5 ,0 ,2 0,0 4 ,0 ,
0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4,java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 92
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 , ,1 ,,0 ,0 ,0 , ,0,,2 ,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 java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 16
7,""!-->>
2, 2, 5, 0, 1 ]
</pre>
<div class="p"><!----></div>
>(2).
Wejava.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 70
<div class="p"><!----></div>
<pre>
&;=java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 51
&62 :=PossibleClassFusions , );
gap62; pi:= Set( fus, map -62; InducedClassFunctionsByFusionMapthe lists java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 69
62; [ TrivialCharacter( tableU ) ], map )[1] );;
gap62; Length( pi );
2
gap62; pi:= Filtered( pi, x -62;ofthe subgroupsfont =+"B</font>2<sup>+24/up>sub><sub.class="oman"Co</span><sub1</ub, 3.span class="roman">Fi</span><sub>24</sub>, and
gap62; Length( pi );
1
gap62; pi7:= pi[1];;
gap62; List( Irr( table2B ), chi -62; ScalarProduct( table2B, chi, pi7 ) );
[ 1, 1, 2, 0, 2, a"sect:>8/a,
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,character of subgroupsof <ontsize"0><font>.
, 4, 1, 4, 2, 6, 1, 4, 0<div
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 1,2,,0,
&62;LoadPackage"atlasrep",false;
2, 3, 1, 0, 2
1,4 4 0,0,0,3,1 ,2 2, 2 0 0,1 2, 3 ,3, 1,2,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
&;()java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
,4,4 ,0, 3, ,1 , 0,0 ,1, 1,5,7 ,
2, 5, 5, 0, 1 ]
<62 SizeScreen 72]);
<""<!----></div>
NextweconsiderU=<spanclass"roman"HN<span>.
<div class="p"><!----></div>
#;tableU= ( "" ;
gap62; fus:= we compute the restriction# 2<size"+0"B/>
gap62; pi:= Set( fus, map -62; InducedClassFunctionsByFusionMap( tableU, table2B,
62; TrivialCharacter( tableU ) ], map )[1] );;
gap&62;Length(pi );
1
gap&62; pi8:= pi[1];
gap&62;List(Irr(table2B ), chi -62; 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 ,6 4,5 6,1 7 4 7 0, ,3,8 2 6 11 ,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,>
10, 7, 19, 7, 7, 8, 10, 14, 18, 19, 5, 10, 12, 23, 7, 12, 6, 24, 6,
4, ,16,8,9 17, 11,24,,26, 21,29 10,18 31
10, 24, 21, 17, 27, 35, 13, 14, 29, 19, 12, 7>> classes5 4096
35, 20, 14, 36, 14, 39, 8, 29, 24, 15, 40, 13on the classes 3 and 7 would differ by 512 times the multiplicity of 4371,
32, 26, 26, 24, 22, 17, 31, 39, 29, 30, 30, 19, 96255a 96256 java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 54
31, 29, 42, 40, [4371 4371,4371, ,275java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
,4,6 10, 10 12,8 12,0 ,2 4 16 , 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,
,, , ,46 58 ,,70 ,104, 56
90 ]
</pre>
classp>!---<>
consider =2< =roman>Fispansub22sub.
There are two [ 8671, 351java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
accordingpre>
One of<ivclass="">!--
of 2.<font size="+0">B</font>, the other is not.
We know that U does not contain the central involution of 2.<font size783,
ence decide java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
<div class="p"><!----></div>
<
gap
# : )java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
gap2;pi:=Set(, -#; InducedClassFunctionsByFusionMap, table2Bjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
gap#;( ;
2
gap62; pi:= Filtered( pi, x -62; java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0
gap62; Length( pi );
1
gap62; pi9:= pi[1];;
gap62; List( Irr( table2B ), chi -62; ScalarProduct)
[ 1, 2, #;( 57477) ;
,10, ,,5,,,14,19,
6, 6, 5, 0, 0, 0,398294
23, 11, 1, 8, 7, 23, 18, 27, 18, 12java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
22 ,33 3,19, , 10,34,12, 12,15,, 17 ,34 34,7, 20 ,
40, 15, 25, 3, 40, 9, 6, 34, 25, 18, 21, 30, 21, java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 16
39, 49, 38, 51, 18 ,, 77 ]
29, 19
75, 29, 24,72,50, 41 72 68,58, 52 , ,64,75 58 69,
65, 49, 85, 75, 75, 63, 68, 65, 63, 90java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 66
90, 109, 109, 2, 3, 6, 9, 8, java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 24
2, 6, 28>-->/>
76, 11, 18, 18, 39, 77, 80, 77, 22, 22, 66, 27, 27, 33, 20, 3< class"oman>Fi/<>24<>sup>&;/>to 3.<span java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 116
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</font></sup> to 2.<font size="+0">B</font>.
<div class="p"><!----></div>
<pre>
gap62; constit:= [ pi1, pi2, pi3, pi4, pi5, pi6, pi7, pi8, pi9 ];;
gap62; pi:= Sum( constit );;
</pre>
<div class="p"><!----></div>
<h2><a ,158 30, 0, 0 ],
4</a> The conjugacy classes of <font size="+0">M</fontjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<a>
<div class="p"><!----></div>
<h3><a name="tth_sEc4.1">
4.1<>
</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>
<
gap62; sizeM ofthepermutation 1sub< size=+"B<font>/>font =0></font></sup>to2.size=+"B/
808017424794512875886459904961710757005754368000000000
gap62; StringPP( sizeM );
"2^46*3^20*5^9*7^6*11^2*13^3*17*19*23*29*31*41*47*59*71"
gap62; sizeM = Sizejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre> </< name""
<div class="p"><!----></div>
For>td> c&8712;</><>G<>a</> />tdalign=right>#124c>Gsuba/sub>/up>le='color: green'>124; </td></tr>
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 inbijection with the corresponding classes in <font size="+0">M</font>,
and that this <tr><td><tt>2B</tt>><td2sup+22<sup.<span class="roman"Cospan><sub>2</ub /td><dalignright"11707448673375/td></r>
<div class="p"><!----></div>
div""<!----></div>
For each element x 8712; <font size="+0">M</font> of order p yle='color: green'>8712; { 2, 3, 5 },
will tableofN<><ontsize=+0>M</font></sub>(9001;x 9002;)
to establish <font size="+0"<tr><td<tt4B</t</td><d2.spanclass/><sub4</>td="ight"1254793905192960000 /<tr>
In order not to count the same class several times,
we proceed<r>td<tt>6A/tt> </td>td2.spanclass"roman"Fispan>sub>2<sub> /d<align=right">64353605265653760000><tr><>
andcollectonlythoseclasses of roots of x for pisthe smallest
prime divisor of the the action is that on the <font size="+0">M</font>-conjugacy class of a,
<div class="p"><!----></div>
For elements x 8712; <font size="+0">M</font> of prime order p &In this,wecompute ninecharacters<>U>sup.font ="0"Bfont<sup>
it is not thata;c
we will use the permutation character values ( java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 31
ad hocjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 21
<div class="p"><!----></div>
<h3><a name="tth_sEc4.2">
4./>&;nbsp; functions<h3<a name="">
</a>
<div class="p"><!----></div>
During the process of finding the conjugacy classes of andthe computation//verification ofthe character tableof 2sup>1+22/up.<pan ="oman>Co<>
we record our knowledgeItwillturn that wedo notneed the character table 2<sup>1+22</sup>.<span class"roman"McL/span>.
in a global <font face="helvetica">GAP</font> variable <ttThe ninecharactersbe in variables
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 &
<dt><b><tt>SizesCentralizers</tt></b></dt>
<dd> <br />
the of java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 59
established up &;]&; #; 2>/> =java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 100
<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 java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 55
each entry is 62map# java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 86
<tt>subtable</tt> (the character table of java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 26
;
unknown positions are unbound).</dd>
</dlgap#;Maximum )
<div class="p"><!----></div>
We initialize this variablejava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 6
and that there is an identity For U = 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>,
<div class="p"><!----></div>
<re>
gap62; head:= rec( Size:= sizeM,
62;
62; fusions:= [],
62; );;
e
62 ()
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</fontjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
and such that <
+0">M</font> whose elements have order divisible bya
that
<""
<pre>
gap62 ExtendTableHeadByRootClasses=function(head ,pos
cents, oldnumber, i,ord;
62;
62; # Initialize the fusion gap&62;pi4=pi[1];
&
62; Add( head62; chi-='color: green'>62;ScalarProduct(table2B chi,pi4 ));;
&
# #Compute ofroot of ''
62; orders:= java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 28
62; p:= orders[ pos ];
62 : )
&62 :=LengthheadOrdersClassRepresentatives;
62;
&;Length )
eady head
#62 iin[1 .. NrConjugacyClassess) ]
62; ord:= orders[i];
62; if ord mod p = 0 and
#; Minimum(PrimeDivisors(ord )=p and
62; (, /pi java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 54
now thatU asubgroupof 2sup222/>.class"oman><<></sub>,
&, cents[i ;
62; Add( head.OrdersClassRepresentatives, orders[i] );
62; java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
62; to 2.<font size="><font>
62; od;
62;
; "#after ", Identifier( s ), ": found ",
# (headO
&62 classes,,now have ",
62; 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
java.lang.StringIndexOutOfBoundsException: Range [43, 3) out of bounds for length 67
> applied g
rootclasses.
That is, we may know just element order <tt>s</tt>
and centralizer order <tt>cent</tt>.
<div class="p"><!----></div>
We are a java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
of a subgroup of <font size="+0">M</font> and the list <tt>poss</tt#;normalsubgroup=centre
in this table which fuse to the class g<java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 26
store this information in the partial class fusion from <tt>s</tt>
that is stored in <tt>head</tt>.
<div class="p"><!----></div>
<prejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
gap62; ExtendTableHeadByCentralizerOrder:= function( head, s, cent, poss )
#62; local ord, fus, i;
62;
62; if IsCharacterTable( s ) then
62; ord:= Set( OrdersClassRepresentatives( s ){ poss } );
62; if Length( ord ) <62; 1 then
&62;;Error( "lasses cannotfuse" );
62; fi;
62; ord:= ord[1];
62; elif IsInt( s ) then
62; ord:= s;
62; fi;
62; Add( head.SizesCentralizers, cent );
62; Add( head.OrdersClassRepresentatives, ord );
62;
62; Print( "#I after order ", ord, " element" );
62; if IsCharacterTable( s ) then
62; # extend the stored fusion from s
62; fus:= First( head.fusions,
62; r -62; Identifier( r.subtable ) = Identifier( s ) );
62; for i in poss do
62; fus.map[i]:= Length( head.SizesCentralizers );
62; od;
62; Print( " from ", Identifier( s ) );
62; fi;
62; Print( ": have ",
62; Length( head.OrdersClassRepresentatives ), " classes\n" );
62; end;;
</pre>
<div class="p"><!----></div>
The permutation character 1<sub>H</sub><sup>G</sup>, where H 8804; G are two groups,
has the property 1<sub>H</sub><sup>G</sup>(g) = 124;C<sub>G</sub>(g)n style='color: green'>124; 183;124;g<sup>G</sup> 8745;H124; / n style='color: green'>124;H124;.
For g 8712; H,
this implies that 124;C<sub>G</sub>(g)124; = 1<sub>H</sub><sup>G</sup>(g) 183;124;H124; / 124;g<sup>G</sup> n style='color: green'>8745;H124;
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>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
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
<>poss/ttlist java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 73
that fuse to g<sup><font size="+0">java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 26
<div&;;
<pre>
gap62; ExtendTableHeadByPermCharValue:= function( head, s, pi_rest_to_s, poss )
62; local pival, cent;
62;
62; pival:= Set( pi_rest_to_s{ poss } );
62; if Length( pival ) <62; 1 then
62; fi;
62;
62; cent:= pival[1] * Size( s ) / Sum( SizesConjugacyClasses( s ){ poss } );
62; ExtendTableHeadByCentralizerOrder( head, s, cent, poss );
62; end;;
</pre>
<div class="p"><!----></div>
<h30 0 3 0 ,,0 , ,,1 ,,,1 , 2 ,0
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>), respectively.
<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 java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 0
as shown in [<a href="#BMverify" name="CITEBMverify">BMW20</a>
in [].
For 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>
in Section <a href="#sect:table_c2b">8</a>.
<div class="p"><!----></div>
huscandetermine < =+"M/>conjugacy classesof elements ofeven order
sfollows
<div class="p"><!----></div>
<>
gap624,12, 11,16,9 7 3,11 13,12 20,5 10 6, ,17,4,
gap62; ClassPositionsOfCentre( s );
,17, 16, 8 9,17 11,12 ,8,24 18 26 21,29,10, , ,
gap62; ExtendTableHeadByRootClasses( head, s, 2 );
#I after 2.B: found 42 classes, now have 43
gap62; s:= CharacterTable( "MN2B" );;
gap62; ClassPositionsOfCentre( s );
,2 ]
head,s ;
#I after 2^1+24.Co1: found 91 classes, now have 134
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h3><a name="tth_sEc4.4">
4.4</a> Classes of elements of java.lang.StringIndexOutOfBoundsException: Range [0, 47) out of bounds for length 40
</a>
<class">--<
We knowdiv="<java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
of order 3,
that normalizershave the structures
3.<span class="roman">Fi</span><sub>24</sub> (for the class <tt>3A</tt>),
3<>+"oman">Suz</span for java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 92
and S<sub>3</sub> ×<span class="roman">Th</span> (for the class <tt>3C</tt>), &62;pi9:=pi1;java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<div class="p"><!----></div>
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 9
the <font face="helvetica">GAP</font> character tables of 3.<span class="roman">Fi</span><sub>24</sub> and <span class="roman">Th</span> are A<font size="-2">TLAS</font> tables
and have been verified, see [<a href="#BMO17" name="CITEBMO17">BMO17</a>].
<div class="p"><!----></div>
We the < size"">/java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 84
that are roots of <tt>3A</tt 39,49,38, 51,18,32,, 19, 42, 41 33 48,64, 27 29, 52,, 38
<div class="p"><!----></div>
<pre>
gap62; s:= CharacterTable( "3.Fi24" );;
gap62; ClassPositionsOfPCore 65,49,8575,75 63, 68,65 ,90,87,83,118,118,74 ,
[ 1, 2 ]
gap62; ExtendTableHeadByRootClasses( head, s, 2 );
#I after 3.F3+.2: found 12 classes, now 0 0,,52, ,,,59,
gap62; s:= CharacterTableDirectProduct( CharacterTable( "Th" ),
62; CharacterTable( "Symmetric"div="p"<!----></div>
gap&62;ClassPositionsOfPCore( s, 3 );
[ 1, 3 ]
gap; ,s ;
#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</span>.2
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 structure
X ×N,
where X has order 3
and N 8773; 3<sup>1+12</sup><sub>+</sub> such that H / N 8773; 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>
gap62; exts:= CharacterTable( "3^(1+12):6.Suz.2" );;
gap62; kernels:= Positions( SizesConjugacyClasses( exts ), 2 );
[ 2, 18, 19, 20 ]
gap62; order3_13:= Filtered( ClassPositionsOfNormalSubgroups( exts ),
62; l -62; Sum( SizesConjugacyClasses( exts ){ l } ) = 3^13 );
[ [ 1 .. 4 ] ]
gap62; 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>
gap62; facts:= List( kernels, i -62; 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 ]" ) ]
gap62; f:= CharacterTable( "2.Suz.2" );;
gap62; facts:= Filtered( facts,
62; x -62; 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>
gap62; kernels:= List( facts,
&; -#;java.lang.StringIndexOutOfBoundsException: Range [60, 58) out of bounds for length 71
[ [ 2 ], [ 2 ] ]
gap&<pre>
gap62; ExtendTableHeadByRootClasses( head, facts[1], 2 );
#I after 3^(1+12):6.Suz.2/[ 1, 19 ]: found 12 classes, now have 165
gap62; ExtendTableHeadByRootClasses( head2, facts[2], 2 );
#I after 3^(1+12):6.div class">
</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""<!----></div>
<pre>
gap62; nams:= RecNames( head );
tives", "izesCentralizers"
" ]
gap&62; ForAll(Difference(nams,["usions ])java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
&62; nam -62;head. ) =head2.( nam ) );
true
gap62; Length( head.fusions );
5
gap62; ForAll( [ 1 .. 4 ], i -62; head.fusions[i] = head2.fusions[i] );
true
gap&;headfusions[]map=.fusions[]map;
true
</pre>
<div class="p"><!----></div>.<a&bsp&U functions/>a =functions"
We continue with establishing the conjugacy classes java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
The question whichofthe two above candidate tables belongs to a subgroup
of <font size="+0">M</font> will record our knowledge about the character knowledgeabout charactertableof< size0>/
<div classwhichis arecordwiththefollowingcomponents.
<h3><a name="tth_sEc4.5">
4.<l =c>
</a>
<div class="p"><!----></div>
The group 2.<font size="+0">B</font> contains two rational conjugacy classes of elementsdt<>ttSizesCentralizers>/<dtjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
of order 5,
<dt>>rdersClassRepresentatives</t<b<java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
<div class="p"><!----></div>
<pre>
gap62; s:= CharacterTable( "2.B" );;
gap62; pos:= Positions( OrdersClassRepresentatives( s ), 5 );
[ 23, 25 ]
gap62; pi{ pos };
[ 1539000, 7875 ]
</pre>
<div class="p"><!----></div>
This establishes two classes <tt>5A< >
of elements of order 5 in <font size="+0">M</font>,
with="roman">HN</span>&124 and 5<sup7</sup> 124;"roman"J/>sub>2/sub>&124,respectively.
<div class="p"><!----></div><ttsubtable</tt> (the character table of the subgroup)
<pre>
gap62; cents:= List( pos,
62; i -62; pi[i] * Size( s ) / SizesConjugacyClasses( s )[i] );
[ 1365154560000000, 94500000000 ]
gap&62 cents = [5 *Size(CharacterTable( "HN" ) ),
62; 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
</span>.,
and <tt>5Btheobject tthead<tt>java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
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>
gap62; s:= > ExtendTableHeadByRootClasses(head,s, java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
CharacterTable( "(D10xHN
gap62; &62 #Initialize information.
[ 1, 45 ]
gap62; ExtendTableHeadByRootClasses( head, s, 45 );
#I after (D10xHN).2: found 5 classes, now have 170
</pre>
<div class="p"><!----></div>
The #; # theclasses ''
with <font face="helvetica">MAGMA</font>, see Section <a href="#sect:table_N5B">10</a>,
arealloweduse the character table from
the <font face="helvetica">GAP</font> character table java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 35
<div class="p"><!----></div>
<pre>
gap62; s:= CharacterTable( "MN5B" );
CharacterTable( "5^(1+6):2.J2.4" )
gap62; 5core:= ClassPositionsOfPCore;fus.api:LengthSizesCentralizers;
[1. 4 ]
gap62; SizesConjugacyClasses( s ){ 5core };
[ &62 " ( ) : "
gap62; > Length.OrdersClassRepresentatives ) - oldnumber,
#I after 5^(1+6):2.J2.4: found 3 classes, now have 173
</pre>
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<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</java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 17
of order 11 with centralizer order 11 124;M<sub>12</sub>e='color: green'>124;
< class=p"<!----></div>
<pre
gap62; s:= CharacterTable( "2.B" );;
gap62; 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>
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos );
#I after order 11 element from 2.B: have 174 classes
</pre>
<div class="&62 := ((s ) )
<h3><a name="tth_sEc4.7">
4.7</a> &> ord[1]
</a>
<div class="p"><!----></div>
The elements of the orders 17, 19, 23, 31, 47 in <font size &62;java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
Sylow subgroups that appear already in 2.<font; : java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 41
< class""<!----></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>
gap62; s:= CharacterTable( "2.B" );;
gap62; pos:= Positions( OrdersClassRepresentatives( s ), 17 );
[ 118 ]
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos );
#java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap62; pos:= Positions( OrdersClassRepresentatives( s ), 19 );
[ 128 ]
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos );
Iafter 19 element 2.: have 176
</pre>
<div class="p"><!----></div>
For elements g of order p 8712; { 23, 31, 47 },
the classes that contain
the nonidentity powers of g,
[>2.java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 140
The equation 124;C<sub><font size="+0">M</font></sub>(g)e='color: green'>124; = 124;2.<font size="+0">B</font>le='color: green'>124; 183;960;(g) / 124;g<sup><font size="+0">M</font></sup> 8745;2.<font size="+0">B</font>124;
implies
<br clear="all" /><table border="0" width="100%"><tr><td>
table align="cellspacing="0" cellpadding=""<tr><nowrap="nowrap" align="center">
124;N<sub><font size="+0">M</font></sub>(9001;g 9002;)124; = [N<sub><font size="+0">M</font></sub>(9001;g 9002;):C<sub><font size="+0">M</font></sub>(g)] 183;124;2.<font java.lang.StringIndexOutOfBoundsException: Index 189 out of bounds for length 31
</td></tr></table>
Note that either the two(head,s,cent ;
fuse in <font size="+0">M</font> or not;
in the former case,
we have [N<sub><font size="+0">M</font></sub>(9001;g 9002;):C<sub><font size="+0">M</font></sub>(g)] = p8722;1 and
124;g<sup><font size="+0">M</font></sup> 8745;2.<font size="+0">B</font>124; = 2 124;g<sup>2.<font size="+0">B</font></sup>124;,
whereas we have
[N<sub><font size="+0">M</font></sub>(9001;g 9002;):C<sub><font size="+0">M</font></sub>(g)] = (p8722;1)/2 and
124;g</pre
Thus we can compute 124;N<sub><font size="+0">M</font></sub>(tyle='color: green'>9001;g 9002;)124; 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>
gap62; s:= CharacterTable( "2.B" );
CharacterTable( "2.B" )
gap62; ord:= OrdersClassRepresentatives( s );;
gap62; classes:= SizesConjugacyClasses( s );;
gap62; p:= 23;;
gap62; pos:= Positions( ord, p );
[ 147, 149 ]
gap62; n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ];
6072
gap62; 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>
fuse in < ="0"></> tojava.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 90
divisible by 2<sup>3</sup>.
We see that this is the case java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
<div class="p"><!----></div>
<pre>
gap62; u:= CharacterTable( "MN2B" );
CharacterTable( "2^1+24.Co1" )
gap62; upos:= Positions( OrdersClassRepresentatives( u ) gap#62;s=CharacterTable( "2.B" );;
[ 289, 294 ]
gap62; java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 12
[ 23, 23 ]
</pre>
<div class="p" [1 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
Thus we have established two classes of element order 23 in <font size="+0">java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 56
<div class="p"><!----></div>
<pre>
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
#I after order 23 element from 2.B: .<a&n >
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } );
#I after order 23 element from 2.B: have 178 java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
</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>
gap62; p:= 31;;
gap62; pos:= Positions( OrdersClassRepresentatives( s ), p );
[ 190, 192 ]
gap62; n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ];
2790
gap62; Collected( Factors( n ) );
[ [ 2, 1 ], [ 3, 2 ], [ 5, 1 ], [ 31, 1 ] ]
gap62; SizesCentralizers( s ){ pos };
[ 62, 62 ]
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
#I after order 31 element from 2.B: have 179 classes
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } );
#I after order 31 It will turn out that eachjava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 66
</pre>
<div class="p"><!----></div>
Finally, we deal with p = 47.
<div class="p"> gap#;facts:=Filtered java.lang.StringIndexOutOfBoundsException: Range [71, 37) out of bounds for length 71
<pre>
gap62; p:= 47;;
gap62; pos:= Positions( OrdersClassRepresentatives <3>a name=tth_sEc45>
[
gap62; n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ];
2162
gap62; Collected( Factors( n ) );
2,
gap62; SizesCentralizers( s ){ pos };
[ 94, 94 ]
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
#I after order 47 element from 2. gap2 pi{pos}java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
gap62; 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>
gap62; p:= 13;;
gap62; pos:= CharacterTable( "(D10xHN).2" )
[ 97 ]
gap62; c:= Size( s ) * pi[ pos[1] ] / classes[ pos[1] ];
73008
gap62; Factors( c );
[ 2, 2, 2, 2, 3, 3, 3, 13, 13 ]
gap62; 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> 183;2<sup>3</sup> 183;3.
<div class="p"><!----></div>
<pre>
gap62; c2b:= CharacterTable( "MN2B" );;
gap62; pos:= Positions( OrdersClassRepresentatives( c2b ), 13 );
[ 220 ]
gap62; 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 order 29,
with centralizer order 3 183;29.
<div class="p"><!----></div>
<pre>
gap62; u:= CharacterTable( "3.Fi24" );;
gap62; pos:= Positions( OrdersClassRepresentatives( u ), 29 );
[ 142 ]
gap62; 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>
gap62; poss:= PositionsProperty( head.SizesCentralizers,
62; x -62; x mod 29 = 0 );
[ 1, 135, 144, 145 ]
gap62; head.OrdersClassRepresentatives{ poss };
[ 1, 3, 87, 87 ]
gap62; 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>
gap62; candprimes:= Difference( PrimeDivisors( head.Size ),
62; [ 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
3 183;29 183;7<sup>i</sup> le='color: green'>183;41<sup>j</sup> 183;59<sup>k</sup> e='color: green'>183;71<sup>l</sup>,
with 0 8804; i 8804; 6 and j, k, l n style='color: green'>8712; { 0, 1 }.
<div class="p"><!----></div>
<pre>
gap62; parts:= Filtered( Collected( Factors( head.Size ) ),
62; x -62; x[1] in candprimes );
[ [ 7, 6 ], [ 41, 1 ], [ 59, 1 ], [ 71, 1 ] ]
gap62; poss:= List( parts, l -62; List( [ 0 .. l[2] ], i -62; l[1]^i ) );;
gap62; cart:= Cartesian( poss );;
gap62; possord:= 3 * 29 * List( cart, Product );;
</pre>
<div class="p"><!----></div>
Only 3 183;29 and 3 183;29 yle='color: green'>183;59 satisfy Sylow's theorem,
that is, 124;<font size="+0">M</font>124; / 124;N<sub><font size="+0">M</font></sub>(9001;x 9002;)124; n style='color: green'>8801; 1 mod 29.
Note that we have [N<sub><font size="+0">M</font></sub>(9001;x style='color: green'>9002;):C<sub><font size="+0">M</font></sub>(x)] = 28.
<div class="p"><!----></div>
<pre>
gap62; good:= Filtered( possord,
62; x -62; ( head.Size / ( 28 * x ) ) mod 29 = 1 );
[ 87, 5133 ]
gap62; List( good, Factors );
[ [ 3, 29 ], [ 3, 29, 59 ] ]
</pre>
<div class="p"><!----></div>
Now we can exclude the possible centralizer order 3 183;29 style='color: green'>183;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>
gap62; Filtered( DivisorsInt( 5133 ), x -62; 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 3 183;29.
<div class="p"><!----></div>
<pre>
gap62; ExtendTableHeadByCentralizerOrder( head, u, 3 * 29, pos );
#I after order 29 element from 3.F3+.2: have 185 classes
<>
<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>
<size+>< a <>8sup><></ubsup&8722<sup>3.
The fact div"p"><!----></div>
by an element of order 4 can be read off from the factor group
O<sub>8</sub><sup>8722;</sup>(3).
<div class="p"><!----></div>
<pre>
gap62; t:= CharacterTable( "O8-(3)" );
CharacterTable( "O8-(3)" )
gap#;Length(Positions(OrdersClassRepresentatives , 41 ))
10
</pre>
< classp>!--</div>
By the above arguments, the java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 10
124;N<sub><font size="+0">M</font></sub>(9001;x 9002;)124;/41 are 5, 7, 59, 71,
where 5 cannot divide the By the arguments in [<a href=" =>/>]java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
As in the case of p = 29, we apply Sylow's theorem,
and get
124;N<sub><font size="+0">M</font></sub>(9001;x 9002;)124; e='color: green'>8712; { java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="p"><!----></div>
p>
gap62; possord:= 2^2 * 41 * DivisorsInt( 2 * 5 * 7^6 * 59 * 71 );;
#;good: (,
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 1640, 163016 ]
gap62; List( good, Factors );
[ [ 2, 2, 2, 5, 41 ], [ 2, 2, 2, 7, 41, 71 ] ]
Telements orders 17,,23 31, in< size="+0">M</>lie in
<div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
Suppose that 71 divides 124;N<sub><font size=" order17and 19are 2<font ="0"B<font>
Then the 71 Sylow subgroup of N<sub><font size="+0">M</font></sub>(9001;x 9002;) is normal
thus normalized by an element of order 8,
and thus centralized by an involution,
<div class="p"><!----> gap2; pos Positions OrdersClassRepresentatives )java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
<pre>
gap62; Filtered( DivisorsInt( good[2] ), x -62; x mod 71 = 1 );
[ 1 ]
</pre>
<div class="p"><!----></div>
Thus we have established a rational class of self-centralizing elements
of order 41.
<div class="p"><!----></div>
p>
gap62; ExtendTableHeadByCentralizerOrder( head, 41, 41, fail );
#I after order 41 element: have 186 classes
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc4.11">
4.11</a> Classes of elements of order divisible by 59</h3><a the group 2.<font size="+0">B</font> contains exactly two Galois conjugate classes+>/>java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 95
/>
d java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
By the above arguments,
the normalizer order of an element of order 59 divides
58 183;7<sup>6</sup> 183;59 yle='color: green'>183;71.
Sylow's theorem admits just the normalizer order 59 ·29.
d =<--<
<pre>
gap62; possord:= 59 * Note that twoof 2< +"<java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 85
gap62; good:= Filtered( possord,
62; x -62; ( head.Size / x ) mod 59 = 1 );
[ 1711 ]
gap# ,)
[ [ 29, 59 ] ]
</pre>
<div class="p"><!----></div>
Thus we have established a pair of Galois conjugate classes of
self-centralizing elements of order 59.
<div class="p"><!----></div>
<pre>
gap62; ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
#I after order 59 element: have 187 classes
gap62; ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
#I after order 59 element: have 188 classes
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc4.12">
4.12</a> Classes of elements of order divisible by 71</h3><a name="elements_71">
</a>
<div class="p"><!----></div>
By the above arguments,
the normalizer order of an element of order 71 divides
70 183;7<sup>5</sup> 183;71.
Sylow' theoremadmits just the normalizer order 71 ·35.
<div class=" we 23java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<pre>
gap62; possord:= 71 * DivisorsInt( 70*7^5 );;
gap62; good:= Filtered( possord,
62; x -62; ( head.Size / x ) mod 71 = 1 );
[ 2485 ]
gap62; List( good, Factors );
[ [ 5, 7, 71 ] ]
</pre>
<div class="p"><!----></div>
&62;=SizesConjugacyClasses ;java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
self-centralizing elements of order 71.
<div class="p"><!----></div>
<pre>
java.lang.StringIndexOutOfBoundsException: Range [39, 7) out of bounds for length 69
#I after order 71 element: have 189 classes
gap62; ExtendTableHeadByCentralizerOrder( head, 71, 71, fail );
#I after order 71 element: have 190 classes
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc4.13">
4.13</a> Classes of<pre>
</a>
<div class="p"><!----></div>
The subgroup 2.<font size="+0">B</font> yields a rational class <tt>7A</tt>
with centralizer order 7 183;124;<span class="roman">He</span>124;.
<div class="p"><!----></div>
<pre>
gap62; s:= CharacterTable( "2.B" );;
gap#62;pos= Positions( OrdersClassRepresentatives( s ), 7 )
[ 41 ]
gap62; ExtendTableHeadByPermCharValue( head, s, pi, pos )e thatthis isthe already the<t2<tt centralizer<fontsize"0>M</>
#I after order 7 element from 2.<ivclass=p>!--<divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
gap62; Last( head.SizesCentralizers ) = 7 * Size( CharacterTable( "He" ) );
true
</pre>
<div class="p"><!----></div>
By additional arguments, we find that the <tt>7A</tt> centralizer has the
structure 7 ×<span class="roman">He</span>,
and the normalizer has the structure (7:3 ×<span class="roman">He</span>).2,
a subdirect product of 7:6 and <span class="roman">He</span>.2.
<div class="p"><!----></div>
Since <span class="roman">He</span> has a pair of Galois conjugate classes of element order 17,
we get also a pair of Galois conjugate classes of element order
7 183;17 = 119.
<div class="p"><!----></div>
<pre>
gap62; ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
#I after order 119 element: have 192 classes
gap62; ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
#I after order 119 element: have 193 classes
</pre>
<div class="p"><!----></div>
The second class of elements of order 7, <tt>7B</tt>,
is established by the fact that the subgroup 3.<span
two classes of elements of order 7, with different values of the
degreee 196 883 character 967; of <font size="+0">M</font>.
<div class="p"><!----></div>
<pre>
gap62; u:= CharacterTable( "3.Fi24" );;
gap62; cand:= Filtered( Irr( u ), x -62; x[1] <= 196883 );;
gap62; rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] } );;
gap62; pos:= <div class="p"><!----></div
]
gap62; rest{ pos };
[ 50, 1 ]
</pre>
<div class="p"><!----></div>
Note that class 43 fuses to <tt>7B</tt> because the restriction of 967;
to 2.<font size="+0">B</font> has the value 50 on <tt>7A</tt>.
<div class="p"><<pre>
<pre>
gap62; ExtendTableHeadByCentralizerOrder( head, u, 7^5 * Factorial(7), [ 43 ] );
#I after order 7 element from 3.F3+.2: have 194 classes
</pre>
<div class="p"><!----></div>
Now the sum of gap&62;pos=Positions((OrdersClassRepresentatives( s ), p );
<div class="p"><!----></div>
<pre>
gap62; Sum( head.SizesCentralizers, x -62; head.Size / x ) = head.Size;
true
</pre>
<div class="p"><!----></div>
We initialize the character table head of <font size="+0">M</font>.
<div class="p"><!----></div>
<pre>
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
62; UnderlyingCharacteristic:= 0,
62; Size:= head.Size,
62; SizesCentralizers:= head.SizesCentralizers,
62; OrdersClassRepresentatives#;p=47;
62; ) );;
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc5">
5</a> The power maps of <font size="+0">M</font></h2><a name="Mpowermaps">
</a>
<div class="p"><!----></div>
Using the element orders of the class representatives of the table head
of <font size="+0">M</font>, and the partial class fusions from the subgroups used in the
previous sections, we compute approximations of the p-th power maps,
for primes p up to the maximal element order in <font size="+0">M</font>.
<div class="p"><!----></div>
Note that we have not yet determined which of the two possible character
tables of the <tt>3B</tt> normalizer belongs to a subgroup of <font size="+0">M</font>,
thus we exclude the corresponding partial fusion.
d
<pre>
gap62; safe_fusions:= Filtered( head.fusions,
2; -#;notIsIdenticalObj(r. facts[1 ;;
gap62; Length( safe_fusions );
gap2; ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
</pre>
<>
First we initialize the <div class="p"><!----></div>
the classes as given by the partial fusions which we have stored.
<div class="p"><!----></div>
<pre>
gap62; for r in safe_fusionsusethejava.lang.StringIndexOutOfBoundsException: Range [36, 32) out of bounds for length 63
62; fus:= InitFusion( r.subtable, m );
62; for i in [ 1 .. Length( r.map ) ] do
if(rmap[ java.lang.StringIndexOutOfBoundsException: Range [44, 45) out of bounds for length 44
&62 if ( fus[i )
62; if fus[i] <62; r.map[i] then
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
62; fi;
62; elif IsInt( r.map[i] ) then
62; if not r.map[i] in fus[i] then
62; Error( "fusion problem" );
62; fi;
62; else
62; if not IsSubset( fus[i], r.map[i] ) then
62; Error( "fusion problem" );
62; fi;
62; fi;
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
62; fi;
62; od;
62; r.fus:= fus;
62; od;
</pre>
<div class="p"><!----></div>
Next we initialize with centralizer order 3 183;29.
and
of the subgroups w. r. t. the current knowledge of the class fusions.
Note that also the knowledge about the class fusions increases this way.
<div class="p"><!----></div>
<pre>
&62 SizesCentralizers( u ) pos };
119
gap62; powermaps:= [];;
gap62; primes:= Filtered( [ 1 .. maxorder ], IsPrimeInt );;
gap62; for p in primes do
62; powermaps[p]:= InitPowerMap( m, p );
62; for r in safe_fusions do
62; subpowermap:= PowerMap( r.subtable, p );
&62 if TransferDiagram subpowermap,rjava.lang.StringIndexOutOfBoundsException: Range [83, 55) out of bounds for length 83
62; Error( "inconsistency" );
62; fi;
62; od;
62; od;
</pre>
<div class="p"><!----></div>
We conditionsuntil further
improvements are found.
<div class="p"><!----></div>
<pre>
gap62; found:= true;;
gap62; res:= "dummy";; # avoid a syntax warning
gap62; while found do
62; Print( "#I start a round\n" );
62; found:= false;
62; for p in primes do
62; for r in safe_fusions do
62; subpowermap:= PowerMap( r.subtable, p );
62; res:= TransferDiagram( subpowermap, r.fus, powermaps[p] );
62; if res = fail then
62; Error( "inconsistency" );
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
62; found:= true;
62; fi;
62; od;
62; od;
62; <div class="p"><div>
#I start a round
# a
62; x-# x[]incandprimes ;
#I start a round
<java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<div class="p"><!----></div>
Let us 318329and3 #;9 #183;59 satisfy Sylow's theorem,
startingwith 5-power .
d =p"<!--></iv>
<pre>
gap62; pos:= PositionsProperty( powermaps[5], IsList );
<re>
gap62; head.OrdersClassRepresentatives{ pos };
[ 15, 15, 39, 39, 59, 59, 71, 71, 119, 119 ]
</pre>
<div class="p"><!----></div>
The ambiguitiesdiv class"p">!--<div
are understandable:
For each of these element orders, there is a pair of Galois conjugate
classes, and the subgroups order, contradiction
these elements.
<div class="p"><!
For each of the primes l 8712; { 59, 71 },
the field of l-th roots of unity contains a unique quadratic subfield,
which is <font size="+0">Q</font>(8730;{8722;l}),
hep- power ,for p coprime to ljava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
Thus have java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 66
the Galois automorphism that raises l-th roots of unity to the p-th power
fixes 8730;{8722;l}.
<div class="p"><!----></div>
In the case of element order l = 119 = 7 183;17,
field -throotsof contains the java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 72
<font size="+0">Q</font>(&>
In order to decide which of them actually occurs,
we look at a subgroup that contains elements of order 119.
The <tt>7A</tt> centralizer in <font size="+0">M</font> has the structure 7 ×<span class="roman">He</span>,
and the normalizer has the structure (7:3 ×<span class="roman">He</span>).2,
a subdirect product of 7:6 and <span class="roman">He</span>.2,
see Section <a href="#elements_7">4.13</a>.
<div class="p"><!----></div>
The classes of element order 119 in the normalizer correspond to
the classes of this element order in <font size="+0">M</font>,
and the character values in the subgroup lie in the field <font size="+0">Q</font>(8730;{8722;119}).
<div class="p"><!----></div>
<pre>
gap#62; u=CharacterTable( "(7:3xHe):2" );;
gap62; ConstructionInfoCharacterTable( u );
["ConstructIndexTwoSubdirectProduct", "7:3", "7:6", "He", "He.2",
[ 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129,
0,131 132,133, 134,208,209 210,211 212
213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224,
225, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307,
308, 309, 310, 311, 312, 313, 314, 315 ], (), () ]
gap62; pos:= Positions( OrdersClassRepresentatives( u ), 119 );
[ 52, 53 ]
gap62; f:= Field( Rationals, List( Irr( u ), x -62; x[pos[1]] ) );;
gap62; Sqrt(-119) in f;
true
</By th above arguments,the possible prime java.lang.StringIndexOutOfBoundsException: Range [63, 64) out of bounds for length 63
<div class="p"><!----></div>
We insert the relevant power map values.
<div class="p"><!----></div>
<pre>
&in [59 , 119 ]do
62; val:= Sqrt( -l );
62; possjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
62; for p in primes do
62; if Gcd( l, p ) = 1 then
62; if GaloisCyc( val, p ) = val then
62; powermaps[p]{ poss }:= poss;
62; gap#62;good: Filtered
62; powermaps[p]{ poss }:= Reversed( poss );
62; [1640 163016]
62; fi;
&62; odjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
62; od;
</pre>
<div class="p"><!----></div>
Now p-th power maps, for p 8805; 17,
are determined uniquely except for the images of two classes
of element order 39.
These classes had been found as roots of <tt>3B</tt>Then the 71 Sylow subgroup of <font =+0"M/font<sub(〈x #9002; is normal
<div class="p"><!----></div>
<pre>
gap62; PositionsProperty( powermaps[17],<pre>
[ 163, 164 ]
gap#;head.OrdersClassRepresentatives{[163 164 ] };
[ 39, 39 ]
gap62; List( Filtered( head.fusions,
62; r -62; IsSubset( r.map, [ 163, 164 ] ) ),
62; r -62; r.subtable );
[ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]
<pre>
<div class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
In order to decide whether the p-th power map fixes or swaps
the two classes, we consider elements of order 78,
which are the square roots of the order 39 elements.
There are three classes of element order 78 in <font size="+0">M</font>,
a rational class that powers to <tt>2A</tt>
and a pair of Galois conjugate classes that power to <tt>2B</tt>.
<div class="p"><!----></div>
<pre4.1</ Classes elements of order divisible by 59</h3><a name="elements_59">
gap62; 78pos:= Positions( head.OrdersClassRepresentatives, 78 );
[,132,
gap62; head.fusions[1].subtable;
CharacterTable( "2.B" )
gap62; Intersection( 78pos, head.fusions[1].map );
[ 37 ]
gap62; s58&1837<6<sup #;59&yle='color: green'>183;.
CharacterTable( "2^1+24.Co1" )
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
[ 132, 133 ]
gap62; Positions( head.fusions[2].map, 132 );
[ 342 ]
gap62; Positions( head.fusions[2].map, 133 );
[344 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
gap62; PowerMap( s, 7 )[342];
344
</pre>
<div class="p"><!----></div>
Since
</re>
subgroup of order 3,
these twoclasses fuse to the non-<font size="0"></ont>classes of elements
of order 78,
and their squares are the classes of element order 39 we are interested in.
<div class="p"><!----></div>
<pre>
gap62; poss:= Filtered( head.fusions, r -62; IsSubset( r.map, [ 163, 164 ] ) );;
gap62; List( poss, r -62; r.subtable );
[ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]
gap62; Position( poss[1].map, 163 );
173
gap&
174
gap62; List( facts, s -62; Positions( OrdersClassRepresentatives( s ), 39 ) );
[ [ 173, 174 ], [ 173, 174 ] ]
gap62; List( facts, s -62; PowerMap( s
[ 174, 174 ]
</pre>
<div class="p"><!----></div>
The field of character values on the two classes of <font size="+0">M</font> is equal
to the corresponding field of character values in the <tt>3B</tt> normalizer,
which is <font size="+0">Q</font>(8730;{8722;39}).
(Note that we have not yet decided which of the two candidate tables belong
to the <tt>3B</tt> normalizer,
but we get the same result for both candidates.)
<div class="p"><!----></div>
<pre>
gap62; fields:= List( facts,
62; s -62; Field( Rationals, List( Irr( s ),; List(good,
62; x -62; x[173] ) ) );;
gap62; Length( Set( fields ) );
1
gap62; Sqrt(-39) in fields[1];
true
</pre>
<div class="pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Now we can set the power map values on the two classes.
< class="">!--</ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<pre>
gap62; val:= Sqrt( -39 );;
gap62; poss:= [ 163, 164 ];;
gap62; for p in primes do
62; if Gcd( 39, p ) = 1 then
62; if GaloisCyc( val, p ) = val then
62; powermaps[p]{ poss }:= poss;
62; else
62; powermaps[p]{ poss }:= Reversed( poss );
62; fi;
62; fi;
62; od;
gap62; List( powermaps, Indeterminateness );
[ , 2048, 1536,, 4,, 2,,,, 2,, 9,,,, 1,, 1,,,, 1,,,,,, 1,, 1,,,,,, 1,,
,, 1,, 1,,,, 1,,,,,, 1,,,,,, 1,, 1,,,,,, 1,,,, 1,, 1,,,,,, 1,,,, 1,,
,,,, 1,,,,,,,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1 ]
</pre>
<div class="p"><!----></div>
In the following,
we use the two candidates for the <tt>3B</tt> normalizer table
for answering most of the remaining questions about the power maps.
Again, the answers are equal for both candidate tables.
<div class="p"><!----></div>
First we initialize the class fusion from the first ,we the<t7At;span class="oman><span>java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<div class="p"><!----></div>
<pre>
gap62; r:= First( head.fusions, r -62; IsIdenticalObj( r.subtable, facts[1] ) );;
gap62; fus:= InitFusion( r.subtable, m );;
gap62; for i in [ 1 .. Length( r.map ) ] do
62; if IsBound( r.map[i] ) then
62; if gap> ExtendTableHeadByCentralize( ,119,119,fail)java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
if fusi <&62; rmapi]then
62; Error( "fusion problem" );
62java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
&62; IsInt rmap[])
62; if not r.map[i] in fus[i] then
62; Error( "fusion problem" );
62; fi;
62; else
62; if not IsSubset( fus[i], r.map[i] ) then
62; Error( "fusion problem" );
62; fi;
62; fi;
&62; fus[i]= r.map[i];
62; fi;
62; od;
gap62; r.fus:= fus;;
</pre>
<div class="p"><!----></div>
... and the class fusion from the second candidate table, ...
<div class="p"><!----></div>
<pre>
62;r2=First(head2. -#62 IsIdenticalObj( r.subtable, facts[2] ) );;
gap62; fus2:= InitFusion( r2.subtable, m );;
gap62; for i in [1 .Length( r2.map)]do
62; if IsBound( r2.map[i] ) then
62; if IsInt( gap> ExtendTableHeadByCentralizer ,75*Factorial7)[43 ;
62; if fus2[i] <62; r2.map[i] then
62; Error( "</pre>
62; fi;
62; elif IsInt( r2.map[i] ) then
62; if not r2.map[i] in fus2[i] then
62; Error( "fusion problem" );
62; fi;
62; else
62; if not IsSubset( fus2[i], r2.map[i] ) then
62; Error( "fusion problem" );
62; fi;
62; fi;
62; fus2[i]:= r2.map[i];
62; fi;
62; od;
gap62; r2.fus:= # =,
</pre>
<div class="p"><!----></div>
... then we create an independent copy of the current approximations
of power maps, and apply the consistency conditions for class fusion and
power maps in the two cases.
<ivclass">!-->/div>
<pre>
powermaps2:= StructuralCopy( powermaps ;;
gap&62;s= .ubtable;
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
gap62; for p in primes do
62; if TransferDiagram( PowerMap( s, p ), fus, powermaps[p] ) = fail then
62; Error( "inconsistency" );
62; fi;
62; od;
gap62; s2:= r2.subtable;
CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
gap62; for p in </div>
62; if TransferDiagram( PowerMap( s2, p ), fus2, powermaps2[p] ) = fail then
62; Error( "inconsistency"<re>
#;fi
62; od;
gap62; powermaps = powermaps2;
true
gap62; List( powermaps, Indeterminateness );
[ , 32, 6
1,, 1,,,, 1,,,,,, 1,,,,,, 1,, 1,,,,,, 1,,,, 1,, 1,,,,,, 1,,,, 1,,,,,
, 1,,,,,,,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1 ]
</pre>
<div class="p"><!----></div>
One open question is about the squares of the non-rational classes
of element order 78.
<div class="p"><!----></div>
<pre>
gap62; powermaps[2]{ [ 132, 133 ] };
[ [ 163, 164 ], [ 163, 164 ] ]
gap62; pos78:= List( facts,
62; s -62; Positions( OrdersClassRepresentatives( s ), 78 ) );
[ [ 235, 236 ], [ 235, 236 ] ]
gap62; &62;Error" ;
[ [ 132, 133 ], [ 132, 133 ] ]
gap62; fus2{ [ 235, 236 ] };
[ [ 132, 133 ], [ 132, 133 ] ]
</pre>
<div class="p"><!----></div>
We may identify one class of element order 78 in the <tt>3B</tt> normalizer
with the corresponding# .fusfus=
<div class="p"><!----></div>
<pre>
gap62; fus[235]:= 132;;
gap62; fus2[235]:= 132;;
2]) lt;#62 fail;
true
gap62; TransferDiagram( PowerMap( s2, 2 ), fus2, powermaps2[2] ) <62; fail;
true
gap62; powermaps = powermaps2;
true
( powermaps{[ 2, ]} Indeterminateness);
[ 8, 64 ]
</pre>
<div class="p"><!----></div>
Since also the cubes of the concerned classes of element order 39
are still not determined, #java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
2nd and 2 &62; subpowermap,p ;
<div class=< 62; TransferDiagramsubpowermap, r.fus, powermaps[p] ) = fail then
<pre>
gap62; powermaps[3]{ [ 163, 164 ] };
[ [ 183, 184 ], [ 183, 184 ] ]
&62 od
true
gap62; List( powermaps{ [ 2, 3 ] }, Indeterminateness );
[ 8, 16 ]
/>
<div class="p"><!----></div>
The next open question is about the cubes of elements of order 93.
The The classesp"!-<
have been found inside subgroups
of the type S<sub>3</sub> ×<span class="roman">Th</span>, and they do not occur in other subgroups
we have considered.
Thus we may choose which of them cubes to the first class of element order
31.
<div class="p"><!----></div>
<pre>
gap62; #62; inprimesdo
&&62; : .java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 59
&62; List( poss, r -62; r.)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
[ CharacterTable( "ThxSym(3)" ) ]
gap62; pos93:= Positions( head.OrdersClassRepresentatives, 93 );
[ 152, 153 ]
gap62;
[ [ 179, 180 ], [ 179
#62 3[152]:= ;
# starta round
62; powermaps[3] ) <62; fail;
true
gap62; List( powermaps{ [ 2, 3 ] }, Indeterminateness );
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
</pre>
<div class="p"><!----></div>
The next open question is about the cubes of elements of order 69.
The classes of element order 69 -a pair of Galois conjugate classes-
have beenfound inside subgroups
of the type 3.<span class="roman">Fi</span><sub>24</sub>, and they do not occur in other subgroups
we have considered.
Thus we may choose which of them cubes to the first class of element order
23.
<div class="p"><!----></div>
<pre>
gap62<ivclass=p>!--<div>
62; r -62; 69 in OrdersClassRepresentatives( r.subtable ) );;
gap62; List( possFor each of element orders, there a pair conjugate
[ CharacterTable( "3.F3+.2" ) ]
gap62; pos69:= Positions( head.OrdersClassRepresentatives, 69 );java.lang.StringIndexOutOfBoundsException: Range [15, 5) out of bounds for length 15
[ 142, 143 ]
gap#2;powermaps[3{pos69 ;
[ [ 177, 178 ], [ 177, 178 ] ]
gap62; powermaps[3]{ [ 142, 143 ] }:= [ 177, 178
gap62; TransferDiagram( PowerMap( poss[1].subtable, 3 ), poss[1].fus,
62; powermaps[3] ) <62; fail;
true
gap62; List( powermaps{ [ 2, 3 ] }, threethree subfields,
[ 8, 1 ]
</pre>
<div class="p"><!----></div>
The next open question is about the squares of certain elements of order 46.
There are two pairs a product 76and< "java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
and the 2nd power map is not yet determined for those classes which power
to the class <tt>2B</tt>.
< class="p"!---</>
<pre>
prejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
[ 26, 27, gap62 ( u)java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
gap62; powermaps[2]{ pos46 };
[ 177, 178, [ 177, 178 ], [ 177, 178 ] ]
gap62; powermaps[23]{ pos46 };
,2 44 44]
</pre>
<div class="p"><!----></div>
ejava.lang.StringIndexOutOfBoundsException: Range [71, 72) out of bounds for length 71
two classes of element order 46 that power to <tt>2A</tt>,
and we have not yet distinguished the other two classes of element order 46.
Thus we may set the power map values.
<div class="p"><!----></div>
<pre>
gap62; powermaps[2]{ [ 118, 120 ] }:= [ 177, 178 ];;
gap62; Indeterminateness ( powermaps[2] );
2
</pre&62; Gcd, 1then
<div class="p"><!----></div>
Now just one value is left to be determined,
the square of a class of element order 18 and
<div class="p"><!---java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>
gap62; powermaps[2][78];
[ 155, 156 ]
gap62; head.OrdersClassRepresentatives[78];
18
gap62; head.
3888
</pre>
<div class="p"><!----></div>
There are two classes with this property in <font size="+0">M</font>,
both are roots of the generators of the normal subgroup of [CharacterTable( 3(1+2)6Suz.[1 19])]
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz<
have the same square.
classp<-->/iv>
<pre>
gap62; Filtered( [ 1 .. Length( head.OrdersClassRepresentatives ) ],
62; i -62; head.OrdersClassRepresentatives[i] = thejava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 52
62; head.SizesCentralizers[i] = gap#;78, fusions1m ;
[ 78, 79 ]
gap62; powermaps[3]{ [ 78, 79 ] };
[ 52, 52 ]
gap62; powermaps Positions(head.fusions[2.map 132 )
154
gap62; First( head.fusions, r -62; 154 in r.map ).subtable;
CharacterTable( "3^ gap&62 , 7)342];
gap62; s:= facts[1];
CharacterTable( "3^(1+12):6.Suz</pre>
gap62; pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
62; i -62; OrdersClassRepresentatives( s )[i] = 18 and
java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 71
[ 67, 83 ]
gap62; PowerMap( s, 2 ){ pos18 };
[ 24, 24 ]
gap62; s:= facts[2];
CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap62; pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
&;i#;( i 18
62; gap62 := Filtered(headfusions, r -62 (r.[163,]));
[ 67, 83 ]
&;s {}
gap#;Position(poss1., ;
</pre>
<div class="p"><!----></div>
We set the last missing value,
and improve the approximations of thegap#;Listfacts, -#62 ,7 )173])
by applying the consistency criteria.
<div class="p"><!----></div>
<pre>
gap62; powermaps[2][78]:= powermaps[2][79];;
gap62; for r in safe_fusions do
62; if not TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
#;powermaps ) then
62; Error( "inconsistent!" );
62; fi;
62; od;
gap62; r:= First( head.fusions, r -62; IsIdenticalObj( r.subtable, facts[1] ) );;
gap62; TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
62; &62 x&;x173]));
true
gap62; r2:= First( head2.fusions, r -62; IsIdenticalObj( r.subtable, facts[2] ) );;
gap62; java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 8
62; powermaps );
true
&;SetComputedPowerMaps ,powermaps);
</pre>
<div class="p"><!----></div>
<>namejava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
; &883java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 114
</a>
<div class="p"><!----></div>
of 196; #;
of <font size="+0">M</font> on the classes of 2.<font size="+0">B</font> 2; fi;
<div class="p"><!----></div>
<pre>
gap62,,1,,1,,1,,, ,,, , 1,,,,,, 1,,,, 1,1,,, 1,
gap62; s:= r.subtable;
CharacterTable( "2.B" )
gap62; cand:= Filtered( Irr( s ), x -62; x[1] <= 196883 )< class=p>!--<divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
gap62; List( cand, x -62; x[1] );
[ 1, 4371, 96255, 96256 ]
gap62; rest:= Sum( cand );;
gap62; rest[1];
196883
</pre>
<div class="p"><!----></div>
Thus we know the values of 967; on those classes of <font size="+0">M</font>
that are known as images of the class fusion from 2.<java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 52
This yields 111 out of the 194 character values.
<div class="p"><!----></div>
<pre>
gap62; chi:= [];;
gap62; map:= r.fus;;
gap62; for i in [ 1 .. Length( map ) ] do
62; if #62 [,m[ java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
62; chi[ map[i] ]:= rest[i];
62; fi;
62; od;
gap62; Number( chi );
111
</pre>
<div class="p"><!----></div>
Also the restriction of 967; to 3.<span class="roman">Fi</span><sub>24</sub> is known,
by Section <a href="#natural">2</a>.
This yields 29 more character values.
<div class="p"><!----></div>
<pre>
gap62; r:= head.fusions[3];;
gap62; s:= r.subtable;
CharacterTable( "3.F3+.2" )
gap62; cand:= Filtered( Irr( s ), x -62; x[1] <= 196883 );;
gap62; rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] } );;
gap62; rest[1];
196883
gap62; map:= r.fus;;
gap62; for i in [ 1 .. Length( map ) ] do
62; if IsInt( map[i] ) then
62; if IsBound( chi[ map[i] ] ) and chi[ map[i] ] <62; rest[i] then
62; Error( "inconsistency!" );
62; fi;
62; chi[ map[i] ]:= rest[i];
62; fi;
62; od;
gap62; Number( </re>
140
</pre>
<div class="p"><!----></div>
Now we compute the restriction< =p>!-->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
There are only 13 possible irreducible constituents of this &62; powermaps2: StructuralCopy(powermaps);
We consider the matrix of values of these characters on those classes
for which the class fusion to <font size="+0">M</font> is uniquely known <em>and</em>
the value of 967; on the image class is known.
This matrix has full rank, thus we can directly compute the decomposition
ofthe restriction into irreducibles
and get 41 more character values.
<div class="p"><!----></div>
<pre>
gap62; r:= head.fusions[2];;
gap62; s:= r.subtable &62; fijava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
gap62; cand:= Filtered( Irr( s ), x [, 32,64,1,1,, 1,, 1,,1, ,, ,,, , ,,, ,, | |