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

[Attila Egri-Nagy]

Dear Ron,

Using the SgpDec pacakge (sgpdec.sf.net) you can construct subgroups
of the wreath product (called cascade products).
See Example 5 in http://arxiv.org/abs/1303.0091.

The package is still in development, so please let me know if you need
help installing/using it.

best wishes,
Attila

On Thu, Oct 24, 2013 at 12:07 PM, Sopsku <rrburns at cox.net> wrote:
> 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
>
>
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