Understanding Direct product of subgroups in GAP

direct-productgapgroup-theory

I am trying to obtain a direct product in GAP, but I am not able to understand the output. The ploblem is the following, I have a GL(2,3) group of 48 elements ($2\times 2$) matrices, and I want to make the direct product between two of its subgroups, lets say $C_2 \times D_8$.

Here the elements in $C_2$ are: 

cg2: [ [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ]

and the elements in $D_8$

gd8: [ [ [ -1, 0 ], [ 0, -1 ] ], [ [ 0, -E(4) ], [ -E(4), 0 ] ], 
    [ [ 0, E(4) ], [ E(4), 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
    [ [ -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)-1/2*E(8)^3 ], 
      [ 1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ] ], 
    [ [ -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)+1/2*E(8)^3 ], 
      [ -1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ] ], 
   [ [ 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)-1/2*E(8)^3 ], 
      [ 1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ] ], 
   [[ 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)+1/2*E(8)^3 ], 
      [ -1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ] ] ]

and the direct product:

dp:=DirectProduct(gc2,gd8);
 <matrix group of size 16 with 4 generators>

    Elements(dp);
    [ [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1]], 
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ], 
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ] ],
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], 
       [ 0, 0, -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ], 
       [ 0, 0, -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ] ], 
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1/2*E(8)+1/2*E(8)^3, 
      1/2*E(8)-1/2*E(8)^3 ], [ 0, 0, 1/2*E(8)-1/2*E(8)^3, 
      1/2*E(8)-1/2*E(8)^3 ] ], 
    [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], 
      [ 0, 0, 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ], 
      [ 0, 0, -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ] ], 
   [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], 
      [ 0, 0, 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ], 
      [ 0, 0, 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], 
     [ 0, 0, -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ], 
     [ 0, 0, -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], 
     [ 0, 0, -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ], 
     [ 0, 0, 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], 
     [ 0, 0, 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ], 
     [ 0, 0, -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ] ], 
   [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1/2*E(8)-1/2*E(8)^3, 
     1/2*E(8)-1/2*E(8)^3 ],[ 0, 0, 1/2*E(8)-1/2*E(8)^3, 
     -1/2*E(8)+1/2*E(8)^3 ] ] ]

So, in this point, I am no able to understand these $4\times 4$ matrices output. I know $C_2$ is a normal subgroup of GL(2,3) and $D_8$ is a non abelian group, but the result confuses me. Does anybody has a clue?.
Thanks in advance.

Best Answer

The group $\operatorname{GL}_2(3)$ has no subgroup isomorphic to $C_2 \times D_8$. If you take the normal $C_2$ copy, you won't find any $D_8$ copies intersecting it trivially. E.g.

gap> gl:=GL(2,3);
GL(2,3)
gap> List(List(ConjugacyClassesSubgroups(gl),Representative),x->StructureDescription(x));
[ "1", "C2", "C2", "C3", "C4", "C2 x C2", "S3", "S3", "C6", "Q8", "D8", "C8", "D12", "QD16", "SL(2,3)", "GL(2,3)" ]
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