Jim Caprioli at The Finland Station

I found this picture [ here ]



Before I continue in the book I want to check if Q8 is indeed a subgroup of GL[2,3]. I use GAP for that task of course.

gap> G:=GL(2,3);
GL(2,3)
gap> Size(G);
48
gap> ccl:=ConjugacyClassesSubgroups(G);;
gap> rep:=List(ccl,Representative);;
gap> List(rep,StructureDescription);
[ "1", "C2", "C2", "C3", "C4", "C2 x C2", "S3", "S3", "C6", "Q8", "D8", "C8",
  "D12", "QD16", "SL(2,3)", "GL(2,3)" ]
gap> H:=rep[10];
Q8
gap> Size(H);
8
gap> Elements(H);
[ [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
  [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ],
  [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
  [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ],
  [ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ]
    , [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ],
  [ [ Z(3), Z(3) ], [ Z(3), Z(3)^0 ] ] ]
gap> Elements(GF(3));
[ 0*Z(3), Z(3)^0, Z(3) ]
gap>