Dear GAP Forum,

On Fri, Mar 08, 2002 at 03:51:06PM -0500, Igor Schein wrote:
>
> I'm trying to understand SmallGroup(40,3). Does it have a name? Any
> insight on its properties and similarities to other groups would be
> more than welcome. For example, I believe it's *similar* to
> SmallGroup(104,3), but it doesn't help.
Both these groups can be described as subgroups of GL(3,p^2),
for appopriate p (p=5 or 13), as follows.

Note that they both have centre of order 2.
The quotients over the centres are groups of matrices
of the form
[[a,b],[0,1]], where a<>0 and a,b belong to the field GF(p), where
p=5 in the case of SmallGroup(40,3), and p=13 in the case of
SmallGroup(104,3). Morever, in the latter case a must be of order
dividing 4.

This gives a hint how to describe SmallGroup(40,3) as a subgroup
of GL(3,5^2):
let C=diag(c^2,1,c), where c is an element of order 8 in the
multiplicative group of GF(5^2). Further, let
D=[[1,1,0],[0,1,0],[0,0,1]]. Then <C,D> is of order 40, has
centre of order 2 generated by C^4=diag(1,1,-1), and its
<C,D>/Comm(<C,D>)=Z_8. So it has a lot in common with SmallGroup(40,3).
Using AllSmallGroup(40), and filtering out the groups that
don't have these two properties, we are indeed left with unique
choice. Hence indeed <C,D>=SmallGroup(40,3).

Similarly, one can do the case of SmallGroup(104,3) to arrive
to an embedding of it into GL(3,13^2).

>
> It might not be technically a proper question for this list, but then
> again, how can I describe this group on sci.math or NMBRTHRY other
> than SmallGroup(40,3), and that doesn't mean much to a GAP non-user,
> or should I say non-GAP user?
>
Hopefully, I answered this, too...

HTH,
Dima