[GAP Forum] Maximal Subgroups for O(7,3)

[Joe Bohanon]

Thanks.  I think I see what I was doing wrong.  I'm still a bit 
perplexed as to why taking a random sample of 2- and 3-elements about 
2000 times never produced a group in the other conjugacy class.

Asst. Prof. Dmitrii (Dima) Pasechnik wrote:
> Dear Joe,
> I don't grok your GAP code, but  the GAP generators from
> http://brauer.maths.qmul.ac.uk/Atlas/clas/O73/
> are correct:
>
> enter them into GAP under the names given there:
> b11:=...
> #...
> #and then do
> G1:=Group(b11,b21);;
>  G2:=Group(a11,a21);;
> H1:=Stabilizer(G1,1);;
> h:=GroupHomomorphismByImages(G1,G2,GeneratorsOfGroup(G1),
> GeneratorsOfGroup(G2));;
> gg:=List(GeneratorsOfGroup(H1),x->Image(h,x));;
> OrbitLengths(Group(gg),[1..1080]);
> [ 702, 378 ]
>
> you see that you get different classes (if they were the same, Group(gg) would
> fix a point)
>
> HTH,
> Dima
>
>
> if you know a representative H of G_2(3) in the original generators, a
> representative of the other class can be constructed by applying an
> outer automorphism to H.
>
> Regards,
> Dmitrii
>
> 2008/11/13 Joe Bohanon <jbohanon2 at gmail.com>:
>   
>> Sorry to those of you who get this twice.  I accidentally sent it to the
>> group pub forum first.
>>
>> I'm trying to get the maximal subgroups for O(7,3) and having some trouble.
>>  ATLAS 3.0 does not have them listed, but ATLAS 2.0 does have the shape and
>> there are 7 permutation representations that can be called up by atlasrep.
>>  For each of those seven, I did the following with G set as the smallest
>> permrep
>>
>> H:=Group(AtlasGenerators("O7(3)",i).generators);
>> iso:=IsomorphismGroups(H,G);
>> S:=Stabilizer(H,1);
>>
>> Then I simply ran Image(iso,S) to get the maximals corresponding to the
>> primitive permreps.  However for the two classes of G2(3), this yields
>> conjugate maximal subgroups.
>>
>> In addition, I also tried to take random elements of order 2 and 3 and try
>> to generate a G2(3), and while I was able to create many of them, none of
>> them were out of this one conjugacy class.
>>
>> Am I missing something here?  I don't think there is a mistake anywhere, as
>> G2(3) is listed as having two classes in Kleidman's tables.
>>
>> Thanks
>> Joe
>>     
>
>