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 <[EMAIL PROTECTED]>:
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
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