<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 Monster group</h1>
<body bgcolor="FFFFFF">
<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 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</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>′</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) ≅ 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 tablehead 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>
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>
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>,Lemma 1].
This follows from the following data about 2.<font size="+0">B</font>.
<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>
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 occur as
constituents of the restriction of χ.
<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>
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>
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>
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</sub>.
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> occur.
The following irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub> can occur as
constituents of the restriction of χ.
<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>
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>.<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>
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><font size="+0">B</font></sup>
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>
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><font size="+0">B</font></sup>
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>
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>,
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 <span class="roman">Co</span><sub>2</sub>,
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>
As a second step,
we compute 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> with the <font face="helvetica">GAP</font> function <tt>PermChars</tt>,
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>
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>
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</font></sup>,
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>
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>.
<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="strategy_classes">
</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>
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="+0">M</font></sup>(x)
and ad hoc arguments.
<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>
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>.
<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>), 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 character table of <font size="+0">B</font>
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 character table is described
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</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 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</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 ≅ 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>
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>
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>
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><font size="+0">M</font></sup>.
<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</span><sub>2</sub>|, respectively.
<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> and normalizer
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 been recomputed
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>)<sub>2.<font size="+0">B</font></sub> yields a class of elements
of order 11 with centralizer order 11 |M<sub>12</sub>| in <font size="+0">M</font>.
<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)] = (p−1)/2 holds.
The equation |C<sub><font size="+0">M</font></sub>(g)| = |2.<font size="+0">B</font>| ·π(g) / |g<sup><font size="+0">M</font></sup> ∩2.<font size="+0">B</font>|
implies
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 the latter case.
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>
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>
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 order 29,
with centralizer order 3 ·29.
<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>
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>
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>
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>
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>
We assume that <font size="+0">M</font> contains a subgroup of the structure 3<sup>8</sup>.O<sub>8</sub><sup>−</sup>(3).
The fact that an element x of order 41 in <font size="+0">M</font> is normalized
by an element of order 4 can be read off from the factor group
O<sub>8</sub><sup>−</sup>(3).
<div class="p"><!----></div>
By the above arguments, the only possible odd prime divisors of
|N<sub><font size="+0">M</font></sub>(〈x 〉)|/41 are 5, 7, 59, 71,
where 5 cannot divide the centralizer order.
As in the case of p = 29, we apply Sylow's theorem,
and get
|N<sub><font size="+0">M</font></sub>(〈x 〉)| ∈ { 2<sup>3</sup> ·5 ·41, 2<sup>3</sup> ·7 ·41 ·71 }.
<div class="p"><!----></div>
Suppose that 71 divides |N<sub><font size="+0">M</font></sub>(〈x 〉)|.
Then the 71 Sylow subgroup of N<sub><font size="+0">M</font></sub>(〈x 〉) is normal
thus normalized by an element of order 8,
and thus centralized by an involution, a contradiction.
<div class="p"><!----></div>
<pre>
gap> Filtered( DivisorsInt( good[2] ), x -> 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>
<pre>
gap> 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 name="elements_59">
</a>
<div class="p"><!----></div>
By the above arguments,
the normalizer order of an element of order 59 divides
58 ·7<sup>6</sup> ·59 ·71.
Sylow's theorem admits just the normalizer order 59 ·29.
<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>
gap> ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
#I after order 59 element: have 187 classes
gap> 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 ·7<sup>5</sup> ·71.
Sylow's theorem admits just the normalizer order 71 ·35.
<div class="p"><!----></div>
Thus we have established a pair of Galois conjugate classes of
self-centralizing elements of order 71.
<div class="p"><!----></div>
<pre>
gap> ExtendTableHeadByCentralizerOrder( head, 71, 71, fail );
#I after order 71 element: have 189 classes
gap> 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 elements of order divisible by 7</h3><a name="elements_7">
</a>
<div class="p"><!----></div>
The subgroup 2.<font size="+0">B</font> yields a rational class <tt>7A</tt>
with centralizer order 7 ·|<span class="roman">He</span>|.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "2.B" );;
gap> pos:= Positions( OrdersClassRepresentatives( s ), 7 );
[ 41 ]
gap> ExtendTableHeadByPermCharValue( head, s, pi, pos );
#I after order 7 element from 2.B: have 191 classes
gap> 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 ·17 = 119.
<div class="p"><!----></div>
<pre>
gap> ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
#I after order 119 element: have 192 classes
gap> 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 class="roman">Fi</span><sub>24</sub> contains
two classes of elements of order 7, with different values of the
degreee 196 883 character χ of <font size="+0">M</font>.
<div class="p"><!----></div>
Note that class 43 fuses to <tt>7B</tt> because the restriction of χ
to 2.<font size="+0">B</font> has the value 50 on <tt>7A</tt>.
<div class="p"><!----></div>
<pre>
gap> 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 class lengths in <tt>head</tt> is equal to the order of <font size="+0">M</font>.
<div class="p"><!----></div>
<pre>
gap> Sum( head.SizesCentralizers, x -> head.Size / x ) = head.Size;
true
</pre>
<div class="p"><!----></div>
We initialize the character tablehead of <font size="+0">M</font>.
<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 tablehead
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.
<div class="p"><!----></div>
<pre>
gap> safe_fusions:= Filtered( head.fusions,
> r -> not IsIdenticalObj( r.subtable, facts[1] ) );;
gap> Length( safe_fusions );
6
</pre>
<div class="p"><!----></div>
First we initialize the class fusions, compatible with the definitions of
the classes as given by the partial fusions which we have stored.
<div class="p"><!----></div>
<pre>
gap> for r in safe_fusions do
> fus:= InitFusion( r.subtable, m );
> for i in [ 1 .. Length( r.map ) ] do
> if IsBound( r.map[i] ) then
> if IsInt( fus[i] ) then
> if fus[i] <> r.map[i] then
> Error( "fusion problem" );
> fi;
> elif IsInt( r.map[i] ) then
> if not r.map[i] in fus[i] then
> Error( "fusion problem" );
> fi;
> else
> if not IsSubset( fus[i], r.map[i] ) then
> Error( "fusion problem" );
> fi;
> fi;
> fus[i]:= r.map[i];
> fi;
> od;
> r.fus:= fus;
> od;
</pre>
<div class="p"><!----></div>
Next we initialize approximations of the power maps of the table of <font size="+0">M</font>,
and improve them using the compatibility of these maps with the power maps
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>
gap> maxorder:= Maximum( head.OrdersClassRepresentatives );
119
gap> powermaps:= [];;
gap> primes:= Filtered( [ 1 .. maxorder ], IsPrimeInt );;
gap> for p in primes do
> powermaps[p]:= InitPowerMap( m, p );
> for r in safe_fusions do
> subpowermap:= PowerMap( r.subtable, p );
> if TransferDiagram( subpowermap, r.fus, powermaps[p] ) = fail then
> Error( "inconsistency" );
> fi;
> od;
> od;
</pre>
<div class="p"><!----></div>
We repeat applying the compatibility conditions until no further
improvements are found.
<div class="p"><!----></div>
<pre>
gap> found:= true;;
gap> res:= "dummy";; # avoid a syntax warning
gap> while found do
> Print( "#I start a round\n" );
> found:= false;
> for p in primes do
> for r in safe_fusions do
> subpowermap:= PowerMap( r.subtable, p );
> res:= TransferDiagram( subpowermap, r.fus, powermaps[p] );
> if res = fail then
> Error( "inconsistency" );
> elif ForAny( RecNames( res ), nam -> res.( nam ) <> [] ) then
> found:= true;
> fi;
> od;
> od;
> od;
#I start a round
#I start a round
#I start a round
#I start a round
</pre>
<div class="p"><!----></div>
Let us see where the power maps are still not determined uniquely,
starting with the 5-th power map.
<div class="p"><!----></div>
The ambiguities for the classes of the element orders 59, 71, and 119
are understandable:
For each of these element orders, there is a pair of Galois conjugate
classes, and the subgroups whose class fusions we have used do not contain
these elements.
<div class="p"><!----></div>
For each of the primes l ∈ { 59, 71 },
the field of l-th roots of unity contains a unique quadratic subfield,
which is <font size="+0">Q</font>(√{−l}),
and the p-th power map, for p coprime to l,
fixes a class of element order l if and only if
the Galois automorphism that raises l-th roots of unity to the p-th power
fixes √{−l}.
<div class="p"><!----></div>
In the case of element order l = 119 = 7 ·17,
the field of l-th roots of unity contains the three quadratic subfields,
<font size="+0">Q</font>(√{−7}), <font size="+0">Q</font>(√{17}), and <font size="+0">Q</font>(√{−119}).
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>(√{−119}).
<div class="p"><!----></div>
We insert the relevant power map values.
<div class="p"><!----></div>
<pre>
gap> for l in [ 59, 71, 119 ] do
> val:= Sqrt( -l );
> poss:= Positions( head.OrdersClassRepresentatives, l );
> for p in primes do
> if Gcd( l, p ) = 1 then
> if GaloisCyc( val, p ) = val then
> powermaps[p]{ poss }:= poss;
> else
> powermaps[p]{ poss }:= Reversed( poss );
> fi;
> fi;
> od;
> od;
</pre>
<div class="p"><!----></div>
Now p-th power maps, for p ≥ 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> elements.
<div class="p"><!----></div>
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>
Since the <tt>3B</tt> normalizer in <font size="+0">M</font> contains a pair of Galois conjugate
classes of element order 78 which power to the generators of the normal
subgroup of order 3,
these two classes fuse to the non-rational <font size="+0">M</font>-classes of elements
of order 78,
and their squares are the classes of element order 39 we are interested in.
<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>(√{−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>
gap> fields:= List( facts,
> s -> Field( Rationals, List( Irr( s ),
> x -> x[173] ) ) );;
gap> Length( Set( fields ) );
1
gap> Sqrt(-39) in fields[1];
true
</pre>
<div class="p"><!----></div>
Now we can set the power map values on the two classes.
<div class="p"><!----></div>
<pre>
gap> val:= Sqrt( -39 );;
gap> poss:= [ 163, 164 ];;
gap> for p in primes do
> if Gcd( 39, p ) = 1 then
> if GaloisCyc( val, p ) = val then
> powermaps[p]{ poss }:= poss;
> else
> powermaps[p]{ poss }:= Reversed( poss );
> fi;
> fi;
> od;
gap> 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 candidate table ...
<div class="p"><!----></div>
<pre>
gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
gap> fus:= InitFusion( r.subtable, m );;
gap> for i in [ 1 .. Length( r.map ) ] do
> if IsBound( r.map[i] ) then
> if IsInt( fus[i] ) then
> if fus[i] <> r.map[i] then
> Error( "fusion problem" );
> fi;
> elif IsInt( r.map[i] ) then
> if not r.map[i] in fus[i] then
> Error( "fusion problem" );
> fi;
> else
> if not IsSubset( fus[i], r.map[i] ) then
> Error( "fusion problem" );
> fi;
> fi;
> fus[i]:= r.map[i];
> fi;
> od;
gap> r.fus:= fus;;
</pre>
<div class="p"><!----></div>
... and the class fusion from the second candidate table, ...
<div class="p"><!----></div>
<pre>
gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.subtable, facts[2] ) );;
gap> fus2:= InitFusion( r2.subtable, m );;
gap> for i in [ 1 .. Length( r2.map ) ] do
> if IsBound( r2.map[i] ) then
> if IsInt( fus2[i] ) then
> if fus2[i] <> r2.map[i] then
> Error( "fusion problem" );
> fi;
> elif IsInt( r2.map[i] ) then
> if not r2.map[i] in fus2[i] then
> Error( "fusion problem" );
> fi;
> else
> if not IsSubset( fus2[i], r2.map[i] ) then
> Error( "fusion problem" );
> fi;
> fi;
> fus2[i]:= r2.map[i];
> fi;
> od;
gap> r2.fus:= fus2;;
</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.
<div class="p"><!----></div>
<pre>
gap> powermaps2:= StructuralCopy( powermaps );;
gap> s:= r.subtable;
CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" )
gap> for p in primes do
> if TransferDiagram( PowerMap( s, p ), fus, powermaps[p] ) = fail then
> Error( "inconsistency" );
> fi;
> od;
gap> s2:= r2.subtable;
CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
gap> for p in primes do
> if TransferDiagram( PowerMap( s2, p ), fus2, powermaps2[p] ) = fail then
> Error( "inconsistency" );
> fi;
> od;
gap> powermaps = powermaps2;
true
gap> List( powermaps, Indeterminateness );
[ , 32, 64,, 1,, 1,,,, 1,, 1,,,, 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>
One open question is about the squares of the non-rational classes
of element order 78.
<div class="p"><!----></div>
We may identify one class of element order 78 in the <tt>3B</tt> normalizer
with the corresponding class of <font size="+0">M</font>, and then draw conclusions.
<div class="p"><!----></div>
Since also the cubes of the concerned classes of element order 39
are still not determined, this question is now decided using that the
2nd and the 3rd power map commute.
<div class="p"><!----></div>
The next open question is about the cubes of elements of order 93.
The classes of element order 93 -a pair of Galois conjugate classes-
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>
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 been found 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>
The next open question is about the squares of certain elements of order 46.
There are two pairs of Galois conjugate classes of element order 46,
and the 2nd power map is not yet determined for those classes which power
to the class <tt>2B</tt>.
<div class="p"><!----></div>
We have defined the two classes of element order 23 as squares of those
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>
There are two classes with this property in <font size="+0">M</font>,
both are roots of the generators of the normal subgroup of order 3 in
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2, and the corresponding two classes in this subgroup
have the same square.
<div class="p"><!----></div>
We set the last missing value,
and improve the approximations of the class fusions we have used,
by applying the consistency criteria.
<div class="p"><!----></div>
<pre>
gap> powermaps[2][78]:= powermaps[2][79];;
gap> for r in safe_fusions do
> if not TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
> powermaps ) then
> Error( "inconsistent!" );
> fi;
> od;
gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
gap> TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
> powermaps );
true
gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.subtable, facts[2] ) );;
gap> TestConsistencyMaps( ComputedPowerMaps( r2.subtable ), r2.fus,
> powermaps );
true
gap> SetComputedPowerMaps( m, powermaps );
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc6">
6</a> The degree 196 883 character χ of <font size="+0">M</font></h2><a name="sect:natcharM">
</a>
<div class="p"><!----></div>
We know the values of the irreducible degree 196 883 character χ
of <font size="+0">M</font> on the classes of 2.<font size="+0">B</font>, by Section <a href="#natural">2</a>.
<div class="p"><!----></div>
Thus we know the values of χ on those classes of <font size="+0">M</font>
that are known as images of the class fusion from 2.<font size="+0">B</font>.
This yields 111 out of the 194 character values.
<div class="p"><!----></div>
<pre>
gap> chi:= [];;
gap> map:= r.fus;;
gap> for i in [ 1 .. Length( map ) ] do
> if IsInt( map[i] ) then
> chi[ map[i] ]:= rest[i];
> fi;
> od;
gap> Number( chi );
111
</pre>
<div class="p"><!----></div>
Also the restriction of χ 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>
gap> r:= head.fusions[3];;
gap> s:= r.subtable;
CharacterTable( "3.F3+.2" )
gap> cand:= Filtered( Irr( s ), x -> x[1] <= 196883 );;
gap> rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] } );;
gap> rest[1];
196883
gap> map:= r.fus;;
gap> for i in [ 1 .. Length( map ) ] do
> if IsInt( map[i] ) then
> if IsBound( chi[ map[i] ] ) and chi[ map[i] ] <> rest[i] then
> Error( "inconsistency!" );
> fi;
> chi[ map[i] ]:= rest[i];
> fi;
> od;
gap> Number( chi );
140
</pre>
<div class="p"><!----></div>
Now we compute the restriction of χ to the <tt>2B</tt> normalizer.
There are only 13 possible irreducible constituents of this restriction.
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 χ on the image class is known.
This matrix has full rank, thus we can directly compute the decomposition
of the restriction into irreducibles,
and get 41 more character values.
<div class="p"><!----></div>
<pre>
gap> r:= head.fusions[2];;
gap> s:= r.subtable;
CharacterTable( "2^1+24.Co1" )
gap> cand:= Filtered( Irr( s ), x -> x[1] <= chi[1] );;
gap> map:= r.fus;;
gap> knownpos:= Filtered( [ 1 .. Length( map ) ],
> i -> IsInt( map[i] ) and IsBound( chi[ map[i] ] ) );;
gap> rest:= List( knownpos, i -> chi[ map[i] ] );;
gap> mat:= List( cand, x -> x{ knownpos } );;
gap> Length( mat );
13
gap> RankMat( mat );
13
gap> sol:= SolutionMat( mat, rest );
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
gap> rest:= sol * cand;;
gap> for i in [ 1 .. Length( map ) ] do
> if IsInt( map[i] ) then chi[ map[i] ]:= rest[i]; fi;
> od;
gap> Number( chi );
181
</pre>
<div class="p"><!----></div>
Which values are still missing?
<div class="p"><!----></div>
For g ∈ <font size="+0">M</font>, we have χ(g) ≡ χ(g<sup>p</sup>) mod p
and |χ(g)|<sup>2</sup> < |C<sub><font size="+0">M</font></sub>(g)|.
We apply these conditions.
In all cases except one, the centralizer orders are small enough
for determining the character value uniquely.
<div class="p"><!----></div>
<pre>
gap> for i in missing do
> ord:= head.OrdersClassRepresentatives[i];
> divs:= PrimeDivisors( ord );
> if ForAll( divs, p -> IsBound( chi[ powermaps[p][i] ] ) ) then
> congr:= List( divs, p -> chi[ powermaps[p][i] ] mod p );
> res:= ChineseRem( divs, congr );
> modulus:= Lcm( divs );
> c:= head.SizesCentralizers[i];
> Print( "#I |g| = ", head.OrdersClassRepresentatives[i],
> ", |C_M(g)| = ", c,
> ": value ", res, " modulo ", modulus, "\n" );
> if ( res + 2 * modulus )^2 >= c and ( res - 2 * modulus )^2 >= c then
> cand:= Filtered( res + [ -1 .. 1 ] * modulus, a -> a^2 < c );
> if Length( cand ) = 1 then
> chi[i]:= cand[1];
> fi;
> fi;
> fi;
> od;
#I |g| = 57, |C_M(g)| = 57: value 56 modulo 57
#I |g| = 93, |C_M(g)| = 93: value 92 modulo 93
#I |g| = 93, |C_M(g)| = 93: value 92 modulo 93
#I |g| = 27, |C_M(g)| = 243: value 2 modulo 3
#I |g| = 95, |C_M(g)| = 95: value 0 modulo 95
#I |g| = 95, |C_M(g)| = 95: value 0 modulo 95
#I |g| = 41, |C_M(g)| = 41: value 1 modulo 41
#I |g| = 59, |C_M(g)| = 59: value 0 modulo 59
#I |g| = 59, |C_M(g)| = 59: value 0 modulo 59
#I |g| = 71, |C_M(g)| = 71: value 0 modulo 71
#I |g| = 71, |C_M(g)| = 71: value 0 modulo 71
#I |g| = 119, |C_M(g)| = 119: value 118 modulo 119
#I |g| = 119, |C_M(g)| = 119: value 118 modulo 119
gap> missing:= Filtered( [ 1..194 ], i -> not IsBound( chi[i] ) );
[ 160 ]
</pre>
<div class="p"><!----></div>
The one missing value can be computed from the scalar product
with the trivial character.
<div class="p"><!----></div>
<pre>
gap> diff:= Difference( [ 1 .. NrConjugacyClasses( m ) ], missing );;
gap> classes:= SizesConjugacyClasses( m );;
gap> sum:= Sum( diff, i -> classes[i] * chi[i] );
-6650349175263480459970863415322722279882752000000000
gap> chi[ missing[1] ]:= - sum / classes[ missing[1] ];
2
</pre>
<div class="p"><!----></div>
Now we decide which of the two candidates for the character table of
the <tt>3B</tt> normalizer is the correct one.
For the first candidate, the restriction of χ
cannot be decomposed into irreducibles.
<div class="p"><!----></div>
The character table of the second candidate is equivalent to
the character table that is stored in the <font face="helvetica">GAP</font> character table library.
<h2><a name="tth_sEc7">
7</a> The irreducible characters of <font size="+0">M</font></h2><a name="sect:irreduciblesM">
</a>
<div class="p"><!----></div>
We will not compute the irreducibles of <font size="+0">M</font> from scratch but verify the
irreducibles from the A<font size="-2">TLAS</font> character table of <font size="+0">M</font>,
in the sense that we use the characters printed in the A<font size="-2">TLAS</font> as an "oracle".
For that,
we compute first a bijection between the columns of our character tablehead
and those of the A<font size="-2">TLAS</font> character table of <font size="+0">M</font>.
This is done by using the following invariants:
element orders, centralizer orders,
the values of χ, and the indirection of χ by the 2nd power map.
<div class="p"><!----></div>
In particular, we see that the sets of invariants are equal for the two
tables.
<div class="p"><!----></div>
Note that we cannot get a better choice of invariants,
since there are 22 pairs of Galois conjugate classes in <font size="+0">M</font>,
and our current knowledge does not allow us to distinguish
the classes of each pair.
<div class="p"><!----></div>
Now we compute a permutation that maps the classes of the A<font size="-2">TLAS</font> table
to suitable classes of our tablehead,
permute the irreducibles of the A<font size="-2">TLAS</font> table accordingly,
and create the "oracle" list.
<div class="p"><!----></div>
(The explicit permutation (32,33)(179,180) makes sure that the power maps
of the A<font size="-2">TLAS</font> table and of our table are compatible.)
<div class="p"><!----></div>
In order to prohibit that <font face="helvetica">GAP</font> tries to compute table automorphisms
of our tablehead of <font size="+0">M</font> (which is impossible without knowing the
irreducible characters),
we set a trivial group as value of the attribute
<tt>AutomorphismsOfTable</tt>;
this will be revised as soon as the irreducibles are known.
<div class="p"><!----></div>
<pre>
gap> SetAutomorphismsOfTable( m, Group( () ) );
</pre>
<div class="p"><!----></div>
<div class="p"><!----></div>
First we compute candidates for the class fusion from 2.<font size="+0">B</font>,
starting from the approximation we have already computed.
Before we apply <font face="helvetica">GAP</font>'s criteria for computing possible class fusions,
we decide about the images of four classes of element orders 40 and 44.
<div class="p"><!----></div>
The two classes of element order 40 are a pair of Galois conjugates
in both 2.<font size="+0">B</font> and <font size="+0">M</font>.
Note that their elements are roots of <tt>2B</tt> elements in <font size="+0">M</font>,
and the classes 110 and 111 correspond to non-rational elements
of order 40 in the <tt>2B</tt> normalizer.
<div class="p"><!----></div>
We may freely choose the fusion from the classes 217 and 218 of 2.<font size="+0">B</font>
because there is a table automorphism of 2.<font size="+0">B</font> that swaps exactly these
two classes.
<div class="p"><!----></div>
<pre>
gap> (217,218) in AutomorphismsOfTable( s );
true
gap> r.fus{ [ 217, 218 ] }:= [ 110, 111 ];;
</pre>
<div class="p"><!----></div>
With the same argument,
also the two classes of element order 44 are a pair of Galois conjugates
both in 2.<font size="+0">B</font> and <font size="+0">M</font>.
<div class="p"><!----></div>
There is a table automorphism of 2.<font size="+0">B</font> that swaps three pairs of classes,
where the classes of element order 44 form one pair,
and each of the other two pairs is fused in <font size="+0">M</font>.
Thus we may again freely choose the fusion from 222 and 223.
<div class="p"><!----></div>
Now the remaining open questions about the class fusion from 2.<font size="+0">B</font>
can be answered by <font face="helvetica">GAP</font>'s function <tt>PossibleClassFusions</tt>.
<div class="p"><!----></div>
<pre>
gap> knownirr:= [ TrivialCharacter( m ), chi ];;
gap> poss:= PossibleClassFusions( s, m,
> rec( chars:= knownirr, fusionmap:= r.fus ) );;
gap> List( poss, Indeterminateness );
[ 1 ]
</pre>
<div class="p"><!----></div>
Now we can induce the irreducibles of 2.<font size="+0">B</font> to <font size="+0">M</font>.
<div class="p"><!----></div>
<pre>
gap> induced:= InducedClassFunctionsByFusionMap( s, m, Irr( s ), poss[1] );;
</pre>
<div class="p"><!----></div>
Next we compute candidates for the class fusions from the subgroups
2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>, 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2, and 3.<span class="roman">Fi</span><sub>24</sub>.
Here we enter also the characters of <font size="+0">M</font> obtained by induction from 2.<font size="+0">B</font>,
because their restrictions to the subgroups provide additional conditions.
<div class="p"><!----></div>
The fusion from 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub> is determined uniquely up to automorphisms
of the subgroup table.
We extend the list of known induced characters.
<div class="p"><!----></div>
In order to compute the fusion from 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2,
we have to consider two classes of element order 56 first.
They are a pair of Galois conjugates both in 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 and <font size="+0">M</font>,
and we may freely choose their fusion because there is a table automorphism
of 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 that swaps exactly these two classes.
<div class="p"><!----></div>
Now the class fusion to <font size="+0">M</font> is determined uniquely up to table automorphisms
of the subgroup,
and we extend the list of induced characters.
<div class="p"><!----></div>
The fusion from 3.<span class="roman">Fi</span><sub>24</sub> is determined uniquely up to automorphisms
of the subgroup table.
We extend the list of known induced characters.
<div class="p"><!----></div>
Next we induce the irreducible characters of cyclic subgroups.
<div class="p"><!----></div>
<pre>
gap> Append( induced,
> InducedCyclic( m, [ 2 .. NrConjugacyClasses( m ) ], "all" ) );
</pre>
<div class="p"><!----></div>
Now we reduce the induced characters with the two known irreducibles of <font size="+0">M</font>,
and apply the LLL algorithm to the result of the reduction;
this yields four new irreducibles.
<div class="p"><!----></div>
Now we use the irreducibles of the A<font size="-2">TLAS</font> table of <font size="+0">M</font> as an oracle,
as follows.
Whenever a character from the oracle list belongs to the <font size="+0">Z</font>-lattice
that is spanned by <tt>lll.remainders</tt>
then we regard this character as verified,
since we can compute the coefficients of the <font size="+0">Z</font>-linear combination, form the character, and check that it has indeed norm 1.
<div class="p"><!----></div>
<pre>
gap> mat:= MatScalarProducts( m, oracle, lll.remainders );;
gap> norm:= NormalFormIntMat( mat, 4 );;
gap> rowtrans:= norm.rowtrans;;
gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
gap> one:= IdentityMat( NrConjugacyClasses( m ) );;
gap> for i in [ 2 .. Length( one ) ] do
> extmat:= Concatenation( normal, [ one[i] ] );
> extlen:= Length( extmat );
> extnorm:= NormalFormIntMat( extmat, 4 );
> if extnorm.rank = Length( extnorm.normal ) or
> extnorm.rowtrans[ extlen ][ extlen ] <> 1 then
> coeffs:= fail;
> else
> coeffs:= - extnorm.rowtrans[ extlen ]{ [ 1 .. extnorm.rank ] }
> * rowtrans{ [ 1 .. extnorm.rank ] };
> fi;
> if coeffs <> fail and ForAll( coeffs, IsInt ) then
> # The vector lies in the lattice.
> chi:= coeffs * lll.remainders;
> if not chi in knownirr then
> Add( knownirr, chi );
> fi;
> fi;
> od;
gap> Length( knownirr );
66
gap> Set( knownirr, chi -> ScalarProduct( m, chi, chi ) );
[ 1 ]
</pre>
<div class="p"><!----></div>
We take the generators of the <font size="+0">Z</font>-lattice and some symmetrizations
of the known irreducibles,
reuce them with the known irreducibles,
and apply LLL again.
<div class="p"><!----></div>
Now we are done.
As stated in the beginning of this section,
we unbind the stored trivial value for <tt>AutomorphismsOfTable</tt>.
<div class="p"><!----></div>
<pre>
gap> SetIrr( m, List( knownirr, x -> ClassFunction( m, x ) ) );
gap> ResetFilterObj( m, HasAutomorphismsOfTable );
gap> TransformingPermutationsCharacterTables( m, atlas_m ) <> fail;
true
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc8">
8</a> Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub></h2><a name="sect:table_c2b">
</a>
<div class="p"><!----></div>
The centralizer C of a <tt>2B</tt> element in <font size="+0">M</font>
has the structure 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>,
which can be constructed as follows.
<div class="p"><!----></div>
Consider a subdirect product H of two groups H/X
and H/N,
where X is a cyclic group of order two,
N is an extraspecial group 2<sup>1+24</sup><sub>+</sub>,
H/N is the double cover of <span class="roman">Co</span><sub>1</sub>, and
H/X is an extension of 2<sup>1+24</sup><sub>+</sub> by <span class="roman">Co</span><sub>1</sub>.
<div class="p"><!----></div>
The centre of H is a Klein four group E whose order two subgroups
are X, Y = Z(N), and a third subgroup D.
We have C ≅ H/D.
<div class="p"><!----></div>
<div class="p"><!----></div>
<center>
<a href="ctblm1.png">Figure</a>
<br />Picture Omitted<br />
</center>
<div class="p"><!----></div>
From the 2-local construction of a matrix representation for <font size="+0">M</font>,
we know the following faithful representations,
given by generators that are preimages of standard generators of <span class="roman">Co</span><sub>1</sub>.
<div class="p"><!----></div>
<ul>
<li> A monomial permutation representation of H/E ≅ 2<sup>24</sup>.<span class="roman">Co</span><sub>1</sub>,
of degree 98 280 over the field with three elements,
<div class="p"><!----></div>
</li>
<li>
a matrix representation of H/X ≅ 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>,
of dimension 4 096 over the field with three elements,
<div class="p"><!----></div>
</li>
<li>
a matrix representation of H/N ≅ 2.<span class="roman">Co</span><sub>1</sub>,
of dimension 24 over the field with three elements.
<div class="p"><!----></div>
</li>
</ul>
<div class="p"><!----></div>
<div class="p"><!----></div>
We proceed in the following steps.
<div class="p"><!----></div>
<ul>
<li> First we compute conjugacy class representatives of H/E.
<div class="p"><!----></div>
</li>
<li>
A faithful permutation representation of H/Y on
2 ·196 560 = 393 120 points is obtained by
glueing the permutation generators of H/E
(on 2 ·98 280 = 196 560 points)
and H/N
(the smallest permutation representation of 2.<span class="roman">Co</span><sub>1</sub>,
on 196 560 points) together.
<div class="p"><!----></div>
The character table of this permutation group can be computed
directly with <font face="helvetica">MAGMA</font> in about 16 hours of CPU time.
<div class="p"><!----></div>
</li>
<li>
Starting from the character table of the factor group H/E,
the 3-modular matrix representation of dimension 4 096 of H/X is used
to compute necessary class splittings from H/E to H/X, as follows.
<div class="p"><!----></div>
This representation lifts to characteristic zero because its restriction
to the extraspecial group M/X
is the unique faithful irreducible 3-modular representation of M/X,
and because this representation extends to the full automorphism group
of the extraspecial group.
Thus we can compute the Brauer character values of the representation
on the 3-regular classes of H/X,
and interpret the values as those of an ordinary character ψ, say.
The tensor square ψ<sup>2</sup> belongs to the group H/E,
and the known values of ψ suffice to determine the decomposition
of ψ<sup>2</sup> into irreducibles of H/E,
and thus to compute also the values of ψ<sup>2</sup> on 3-singular classes.
Taking square roots, we get all values of ψ, up to signs.
(We cannot distinguish which of the two values belongs to which of the two
preimage classes, which just means that we are defining these classes
by choosing the positive value for one of them, and the negative value
for the other one.)
<div class="p"><!----></div>
</li>
<li>
From now on, we argue character-theoretically.
<div class="p"><!----></div>
The missing irreducible characters of H/X are computed as
tensor products of ψ with the irreducible characters of
the factor group H/M ≅ <span class="roman">Co</span><sub>1</sub>.
<div class="p"><!----></div>
Note that this procedure yields enough new irreducible characters
such that the sum of degree squares of all known irreducibles of H/X
equals the order of this group.
This implies that there are not more class splittings w. r. t. the
fusion from H/X to H/E than the splittings forced by the values
of ψ.
<div class="p"><!----></div>
</li>
<li>
Using the two class splittings from H/X and H/Y to H/E,
we compute necessary class splittings from H to H/E.
That is, we create a character tablehead for H together with
class fusions to H/X and H/Y,
assuming that not more columns occur than is forced by H/X and H/Y:
For each class of H/E that splits in both H/X and H/Y,
we get four preimage classes in H.
For each class that splits in exactly one of H/X and H/Y,
we get two preimage classes in H.
For each class that splits in none of H/X and H/Y,
we get one preimage class in H.
<div class="p"><!----></div>
Then we take those irreducible characters of H/X and H/Y,
respectively, that do not have E/X or E/Y, respectively,
in their kernel;
we form tensor products of them, which yields characters with kernel D,
and apply the LLL algorithm to them.
<div class="p"><!----></div>
This yields all missing irreducibles of H:
The degree squares of all now known irreducibles sum up to the order
of H, which means that no more class splitting occurs.
<div class="p"><!----></div>
Finally, we compute the power maps (and thus the element orders)
of the character table of H.
The result is a character table that is permutation equivalent to
the character table that is stored in <font face="helvetica">GAP</font>'s table library.
<div class="p"><!----></div>
</li>
</ul>
<div class="p"><!----></div>
<h2><a name="tth_sEc9">
9</a> Appendix: The character table of 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2</h2><a name="sect:norm3B">
</a>
<div class="p"><!----></div>
<h3><a name="tth_sEc9.1">
9.1</a> Overview</h3>
<div class="p"><!----></div>
The <tt>3B</tt> normalizer in <font size="+0">M</font> has the structure 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2.
Its character table has been computed by Richard Barraclough and
Robert A. Wilson, see [<a href="#BW07" name="CITEBW07">BW07</a>].
In order to describe a reproducible construction of the character table
that does not assume the character table of <font size="+0">M</font>,
we recompute this table.
<div class="p"><!----></div>
<div class="p"><!----></div>
Our approach is similar to that in [<a href="#BW07" name="CITEBW07">BW07</a>]:
The subgroup in question is a factor group of the split extension H of
the extraspecial group N = 3<sup>1+12</sup><sub>+</sub> by 6.<span class="roman">Suz</span>.2,
and we compute the character table of this bigger group H.
<div class="p"><!----></div>
<div class="p"><!----></div>
<center> <a href="ctblm2.png">Figure</a>
<br />Picture Omitted<br />
</center>
<div class="p"><!----></div>
We have H / N ≅ 6.<span class="roman">Suz</span>.2.
Let M be the normal subgroup of H above N such that
H / M ≅ 2.<span class="roman">Suz</span>.2 holds.
Then M = X ×N for a normal subgroup X of order 3,
where the extension of M / X by H / M is split.
Let N<sub>2</sub> be the normal subgroup of H above N such that
H / N<sub>2</sub> ≅ 3.<span class="roman">Suz</span>.2 holds.
Then M<sub>2</sub> = M N<sub>2</sub> has the property H / M<sub>2</sub> ≅ <span class="roman">Suz</span>.2.
<div class="p"><!----></div>
Let Y = Z(N), of order 3.
Then E = X ×Y ≅ 3<sup>2</sup> is elementary abelian,
with diagonal normal subgroups D<sub>1</sub> and D<sub>2</sub>.
It will turn out that the extensions of M / D<sub>1</sub> and M / D<sub>2</sub> by H / M
are non-split.
<div class="p"><!----></div>
H / Y is a subdirect product of H / N and
H / E ≅ 3<sup>12</sup>.2.<span class="roman">Suz</span>.2,
we have H / Y ≅ (3 ×3<sup>12</sup>).2.<span class="roman">Suz</span>.2.
<div class="p"><!----></div>
The group H is a subdirect product of H / N and H / X.
<div class="p"><!----></div>
The <tt>3B</tt> normalizer in <font size="+0">M</font> is isomorphic to
one of the two factor groups H / D<sub>1</sub>, H / D<sub>2</sub>.
(The decision which of the two groups occurs as a subgroup of <font size="+0">M</font>
appears in Section <a href="#sect:natcharM">6</a>.)
<div class="p"><!----></div>
We use the following approach to compute the character table of H.
<div class="p"><!----></div>
<ul>
<li> Compute permutation generators <tt>gensHmodX</tt> of H / X,
of degree 3<sup>13</sup>,
see Section <a href="#HmodXpermgens">9.2</a>.
The first two generators are standard generators of 2.<span class="roman">Suz</span>.2,
the third generator lies in M / X.
<div class="p"><!----></div>
</li>
<li>
Fetch permutation generators <tt>gensHmodN</tt> of H / N<sub>2</sub>,
of degree 5 346,
and compute permutation generators <tt>gensH</tt> of H,
of degree 1 599 669 = 1 594 323 + 5 346,
see Section <a href="#Hpermgens">9.3</a>.
The first two generators are standard generators of 6.<span class="roman">Suz</span>.2,
the third generator lies in N.
<div class="p"><!----></div>
</li>
<li>
Let <font face="helvetica">MAGMA</font> compute the character table of H from <tt>gensH</tt>,
see Section <a href="#computeHtable">9.4</a>.
<div class="p"><!----></div>
</li>
</ul>
<div class="p"><!----></div>
<b>Remark:</b>
In an earlier construction (in September 2020),
we asked <font face="helvetica">MAGMA</font> to compute the character table of H / Y,
and then used individual conjugacy tests for first setting up the class fusion
from H to H / Y and then determining the full character tablehead of H.
The computation of the missing irreducible characters of H was then
not difficult, using character theoretic methods.
However, trying the same input files in 2024 showed many cases where
some of the required conjugacy tests did not finish in reasonable time.
Luckily, the automatic computation described in Section <a href="#computeHtable">9.4</a>
worked.
<div class="p"><!----></div>
<h3><a name="tth_sEc9.2">
9.2</a> A permutation representation of H / X</h3><a name="HmodXpermgens">
</a>
<div class="p"><!----></div>
The A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>] contains
a faithful representation of H,
as a group of 38 ×38 matrices over the field with three elements.
The generating matrices are called <tt>M3max7G0-f3r38B0.m1</tt>, ...,
<tt>M3max7G0-f3r38B0.m4</tt>.
These matrices are block diagonal matrices with blocks of the lengths
24 and 14.
<div class="p"><!----></div>
Here we will use just the lower right 14 ×14 blocks,
which generate the factor group H / X ≅ 3<sup>1+12</sup><sub>+</sub>:2.<span class="roman">Suz</span>.2.
(Note that the 12-dimensional irreducible representation of 2.<span class="roman">Suz</span>.2
over the field with 3 elements respects a symplectic form,
and the embedding of 2.<span class="roman">Suz</span>.2 into <span class="roman">Sp</span>(12,3)
-which is the automorphism group of the extraspecial group 3<sup>1+12</sup><sub>+</sub>-
yields a construction of the semidirect product 3<sup>1+12</sup><sub>+</sub>:2.<span class="roman">Suz</span>.2
as a group of 14 ×14 matrices.)
<div class="p"><!----></div>
Later we will construct a faithful permutation representation of H
as a subdirect product of H / X and H / N ≅ 6.<span class="roman">Suz</span>.2.
Let <tt>G</tt> be the group generated by the 14 ×14 matrices.
<div class="p"><!----></div>
When one deals with H / X, the fourth generator is redundant,
we will leave it out in the following.
<div class="p"><!----></div>
We use just the following facts.
The <tt>G</tt>-action on GF(3)<sup>14</sup> has six orbits,
from which we get a faithful permutation representation of H / E ≅ 3<sup>12</sup>.2.<span class="roman">Suz</span>.2 on 196 560 points
and a faithful representation <tt>homHtoHmodX</tt>
of a group of the structure 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 on 1 594 323 points.
Since the image contains a subgroup 2.<span class="roman">Suz</span>.2,
the image is the split extension H / X of 3<sup>1+12</sup><sub>+</sub> by 2.<span class="roman">Suz</span>.2.
<div class="p"><!----></div>
We conclude that the action on 196 560 points represents the group
H / E.
As for the action on 1 594 323 points,
we show that it has a nonabelian normal subgroup of the order 3<sup>13</sup>.
<div class="p"><!----></div>
Next we show that the first two elements from the generating set
are standard generators of 2.<span class="roman">Suz</span>.2.
For that, we first show that these elements are preimages
of standard generators of <span class="roman">Suz</span>.2,
by computing that the words in question lie in the centre of 2.<span class="roman">Suz</span>.2,
and then show that the elements satisfy the conditions of
standard generators of 2.<span class="roman">Suz</span>.2,
that is, the second generator has order 3,
see the page on <span class="roman">Suz</span> in the A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>].
<h3><a name="tth_sEc9.3">
9.3</a> A permutation representation of H</h3><a name="Hpermgens">
</a>
<div class="p"><!----></div>
The group H is a subdirect product of H / X and H / N<sub>2</sub> ≅ 3.<span class="roman">Suz</span>.2,
w.r.t. the common factor group H / M<sub>2</sub> ≅ <span class="roman">Suz</span>.2.
Since we know that our generators for H / X are compatible with
standard generators of the factor group 2.<span class="roman">Suz</span>.2,
it is sufficient to form the diagonal product of our representation of
H / X and a representation of 3.<span class="roman">Suz</span>.2 on standard generators.
<div class="p"><!----></div>
The first two of the generators are standard generators of 6.<span class="roman">Suz</span>.2.
Note that we know already that they are preimages of standard generators
of <span class="roman">Suz</span>.2,
it remains to show that they are elements C, D where D has order 3
and CDCDD has order 7.
<div class="p"><!----></div>
This character table was used in Section <a href="#elements_3">4.4</a>.
<div class="p"><!----></div>
<h2><a name="tth_sEc10">
10</a> Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2</h2><a name="sect:table_N5B">
</a>
<div class="p"><!----></div>
The normalizer of a <tt>5B</tt> element in <font size="+0">M</font> has the structure
5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2,
generators for this group as a permutation group of degree 78 125
are available in the A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>].
<font face="helvetica">MAGMA</font> [<a href="#Magma" name="CITEMagma">BCP97</a>] can compute the character table from the group
within a few minutes,
the result turns out to be equivalent to the table that is available in
<font face="helvetica">GAP</font>'s character table library.
<div class="p"><!----></div>
This character table was used in Section <a href="#elements_5">4.5</a>.
<div class="p"><!----></div>
<h2>References</h2>
<dl compact="compact">
<dt><a href="#CITEMagma" name="Magma">[BCP97]</a></dt><dd>
W. Bosma, J. Cannon, and C. Playoust, <em>The Magma algebra system. I.
The user language</em>, J. Symbolic Comput. <b>24</b> (1997),
no. 3-4, 235-265. MR 1484478
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEAtlas2017" name="Atlas2017">[BGH<sup>+</sup>17]</a></dt><dd>
M. Bhargava, R. Guralnick, G. Hiss, K. Lux, and P. H. Tiep (eds.), <em>Finite
simple groups: thirty years of the Atlas and beyond</em>, Contemporary
Mathematics, vol. 694, Providence, RI, American Mathematical Society, 2017.
MR 3682583
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEBMO17" name="BMO17">[BMO17]</a></dt><dd>
T. Breuer, G. Malle, and E. A. O'Brien, <em>Reliability and reproducibility
of Atlas information</em>, in Bhargava et al. [<a href="#Atlas2017" name="CITEAtlas2017">BGH<sup>+</sup>17</a>],
p. 21-31. MR 3682588
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEBMverify" name="BMverify">[BMW20]</a></dt><dd>
T. Breuer, K. Magaard, and R. A. Wilson, <em>Verification of the ordinary
character table of the Baby Monster</em>, J. Algebra <b>561</b> (2020),
111-130. MR 4135540
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEMverify" name="Mverify">[BMW24]</a></dt><dd>
<br /><table align="left" border="0"><tr><td width="50">
<hr />
</td></tr></table><!--hbox-->
, <em>Verification of the conjugacy classes and ordinary character table of the Monster</em>, submitted, 2024.
<div class="p"><!----></div>
</dd>
<dt>[]</dt><dd>T. Breuer, <em>Constructing the ordinary character tables of some Atlas
groups using character theoretic methods.</em>, <a href="https://export.arxiv.org/abs/1604.00754">arXiv:1604.00754</a>.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEBW07" name="BW07">[BW07]</a></dt><dd>
R. W. Barraclough and R. A. Wilson, <em>The character table of a maximal
subgroup of the Monster</em>, LMS J. Comput. Math. <b>10</b> (2007),
161-175. MR 2308856
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEGAP" name="GAP">[GAP24]</a></dt><dd>
<em><font face="helvetica">GAP</font> - Groups, Algorithms, and Programming,
Version 4.13.1</em>, <a href="https://www.gap-system.org"><tt>https://www.gap<tt>-</tt>system.org</tt></a>, Mar 2024.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEGMS89" name="GMS89">[GMS89]</a></dt><dd>
R. L. Griess Jr., U. Meierfrankenfeld, and Y. Segev, <em>A uniqueness proof
for the Monster</em>, Ann. of Math. (2) <b>130</b> (1989), no. 3,
567-602. MR 1025167
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEAGRv3" name="AGRv3">[WWT<sup>+</sup>]</a></dt><dd>
R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton,
S. Nickerson, S. Linton, J. Bray, and R. Abbott, <em>ATLAS of Finite Group
Representations</em>, <a href="http://brauer.maths.qmul.ac.uk/Atlas/v3"><tt>http://brauer.maths.qmul.ac.uk/</tt>
<tt>Atlas/v3</tt></a>.</dd>
</dl>
<div class="p"><!----></div>
<div class="p"><!----></div>
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