[GAP Forum] D8 as a product of Q=D8/Center and its Center

[Sopsku]

Dear Forum

I am tryng to use GAP to "reconstruct" the group D8 as some sort of product
of its quotient group Q=D8/Center(D8) and its Center(D8).

gap> D8:=DihedralGroup(8); # or however best defined for this
<pc group of size 8 with 3 generators>
gap> ZD8:=Center(D8);
Group([ f3 ])
gap> Q:=D8/ZD8;
Group([ f1, f2, <identity> of ... ])
gap> StructureDescription(Q);
"C2 x C2"

I know it can't be direct product. I reorganized the Cayley Graph for D8 by
grouping the cosets as nodes and and looking at the lines connecting the
nodes. They cross and so a direct product will not do it. Verify this:

gap> d:=DirectProduct(Q,ZD8);
<pc group of size 8 with 3 generators>
gap> StructureDescription(d);
"C2 x C2 x C2"

Not a direct product. See if I can make a semidirect product

gap> A:=AutomorphismGroup(ZD8);
<group of size 1 with 2 generators>
gap> List(A,Order);
[ 1 ]

I don't see how to match up any orders and proceed from here.

I would appreciate any help as how to use GAP functionality to construct 
this product Thank you for any help.
      Ron