[GAP Forum] Equal structure descriptions

[Dima Pasechnik]

On Mon, Aug 20, 2012 at 03:00:45PM +0800, R.E. Boss wrote:
> What can I conclude from the equal structure descriptions below I stumbled upon?
>
> gap> StructureDescription(SmallGroup(64,156));
> "Q8 : Q8"
there is a normal subgroup Q_8 (the group of quaternions),
and the quotient of your group over it
is isomorphic to Q_8, and the extension is split.
> gap> StructureDescription(SmallGroup(64,158));
> "Q8 : Q8"
same as above, but the extension is different.
>
> gap> StructureDescription(SmallGroup(64,155));
> "(C8 : C4) : C2"
there is a normal subgroup C_8:C_4 (a split extension of C_8 by C_4)
on which C_2 acts nontrivially (C_k denotes the cyclic group of order
k).
>
> gap> StructureDescription(SmallGroup(64,157));
> "(C8 : C4) : C2"
as above, but some of the extensions involved are different.
>
> gap> StructureDescription(SmallGroup(64,159));
> "(C8 : C4) : C2"
ditto.
>
> gap> StructureDescription(SmallGroup(64,160));
> "(C2 x C2) . (C2 x D8) = (C4 x C2) . (C2 x C2 x C2)"
there are normal subgroups C_2xC_2 and C_4xC_2.
The whole group can be described in two ways:
1) as a (possibly non-split) extension of C_2xC_2 by
C_2xC_8
1) as a (possibly non-split) extension of C_4xC_2 by
C_2xC_2xC_2.


I hope this gives you an idea...
Best,
Dmitrii


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