[GAP Forum] what does "." and "=" mean in StructureDescription?

[Max Horn]

Dear Will,

> On 20 Jul 2016, at 07:35, Will Chen <oxeimon at gmail.com> wrote:
> 
> A good example is AllSmallGroups(96,191), which seems to have
> StructureDescription:
> 
> "(C2.S4=SL(2.3).C2):C2"
> 
> I understand that the colon ":" denotes semidirect product, so this is the
> semidirect product of something with the cyclic group of order 2.

Correct.

> 
> Here is C2 the normal subgroup of the semidirect product? Does the normal
> subgroup always appear on the right?

No, always on the left, i.e. C_2 acts on the normal subgroup on the left.

> What about the rest? Since the whole group has order 96, the stuff on the
> left of the ":" must have order 48, so I can only assume that the "=" means
> "or equivalently...". Is that correct?

Yes, correct.

> 
> In this case, I'm guessing "." refers to a non-split extension?

Correct.

> If I see G.H, which is the normal subgroup?

The normal subgroup is always on the left. This fits with the overall GAP convention of always considering right actions.


Most of this is explained in the manual (with the exception of the "=" sign, unfortunately). If you enter "?StructureDescription" on the GAP prompt and press return, you can see the manual entry. Alternatively, look here:
<http://www.gap-system.org/Manuals/doc/ref/chap39.html#X87BF1B887C91CA2E>


Cheers,
Max