> < ^ Date: Fri, 03 Apr 1998 11:52:07 +0200 (CEST)
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: Character Tables and Conjugacy Classes of Maximal Subgroups of G2(3)

Dear GAP Forum,

Bernardo Rodrigues wrote

I have been trying to find the Character Tables and the Conjugacy
Classes of the Maximal Subgroups of the Chevalley Group G2(3), and
was unsuccessful. Could anyone point out what I should do?
Actually, I am looking for the Character Table and the Conjugacy Classes
of the group "2^3.GL(3,2)" which is the 8th maximal subgroup of G2(3).

Unfortunately the GAP library of character tables does not contain
the character table of the 8th maximal subgroup of G2(3).
But this table can be constructed easily by the command

```tbl:= CharTable( PermGroup( PerfectGroup( 1344, 2 ) ) );
```

(Note that the group in question is perfect, and that there are exactly
two perfect groups of order 1344, which can be fetched from the library
of perfect groups with `PerfectGroup'.
The first of these groups is a split extension of 2^3 with L3(2),
the second is isomorphic to the required maximal subgroup of G2(3).
The `PermGroup' command is used only to speed up the computations.)

Alternatively, one can construct the table from a library table
by factoring out a normal subgroup.
Namely,

```tbl:= CharTable( "4^3.L3(2)" ) / [ 1, 2 ];
```

yields the desired character table, too.

The fusion of conjugacy classes of `tbl' into the character table
of G2(3) is not uniquely determined by the tables.
One can compute all possible fusions with

```SubgroupFusions( tbl, CharTable( "G2(3)" ) );
```

and then choose a fusion that has the class 3E of G2(3) in its image.
In the ATLAS table of G2(3), which is equal to `CharTable( "G2(3)" )',
this is the class at position 7.
There are two fusions with this property, they are equal modulo the
action of the group automorphism of G2(3).

I hope this helps.

Thomas Breuer

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