[GAP Forum] ConjugacyClasses

[Stefan Witzel]

Hi gap forum,

can anyone explain the following behavior to me (saying that PSL(3,2) has to distinct conjugacy classes that are conjugate)?

vec1:=[1,0,0]*Z(2)^0;;
sl:=SL(3,2);;
orb:=Orbit(sl,vec1,OnLines);;
act:=ActionHomomorphism(sl,orb,OnLines);;
psl:=Image(act);;
ConjugacyClasses(psl); 

[ ()^G, (3,4)(6,7)^G, (2,3,5,4)(6,7)^G, (2,3,6)(4,7,5)^G, (1,2,3,4,7,5,6)^G, (1,2,3,5,6,7,4)^G ]

g:=Representative(ConjugacyClasses(psl)[5]);;
h:=Representative(ConjugacyClasses(psl)[6]);;
c:=(2,6,4)(3,5,7);;
c*g^(-1)*c^(-1)=h;

true

IsSubgroup(psl,Group(c));

true

I guess one could as well use PSL(3,2) out of the box but that gives a different permutation presentation and I didn't want to work out c anew. I also know that PSL(3,2)=SL(3,2)=GL(3,2) etc., but that's not the point of the question. (I do not know what would be the right way to say "IsElement(psl,c)" in the end, so I would be interested in that.)
Thanks for the help!

Best regards,
Stefan

P.S.: In case you are wondering, I'm using GAP version 4.4.12.