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

[Asst. Prof. Dmitrii (Dima) Pasechnik]

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