[GAP Forum] Extension

[Vipul Naik]

This is probably not the most efficient way, but here's one way you
can do it:

gap> G := PSL(2,25);;
gap> A := AutomorphismGroup(G);;
gap> M := SubgroupsOfIndexTwo(A);;

M will be a list of the three groups you want; you can access these as
M[1], M[2], and M[3].

Basically, G is PSL(2,25) and its automorphism group is PGammaL(2,25),
which is defined as the semidirect product of PGL(2,25) and the Galois
group of the field of 25 elements over the field of 5 elements. The
quotient PGammaL(2,25)/PSL(2,25) has order 4 and is a Klein
four-group. The three subgroups of order two and index two in there
correspond to the three groups of index two in PGammaL(2,25). One of
these will be PGL(2,25), the other will be the semidirect product of
PSL(2,25) with the Galois group, and there will be a third one.

Unfortunately, as far as I can make out, the SubgroupsOfIndexTwo takes
a few minutes to run for groups of this size, so you will need to be
patient.

Vipul

* Quoting sadegh salehi who at 2011-11-30 06:25:40+0000 (Wed) wrote
> Hello dear GAP Forum members,
>  
> By the Atlas of finite groups, we know that the simple group PSL(2,9)has three cyclicautomorphic extensions:  S6 (Symmetric group of degree 6), PGL(2,9) and the 'Mathieu' group M(10).
> Also we note that M(10) is non-split extension of PSL(2,9). So it can not be Created by the semidirect product.
> Is there any way to access the automorphic extension of PSL(2,25) or PSL(2,49) , PSL(3,4), ...  in Gap?
> I will be appreciate for any suggestion.
> 
> With Best Regards.
> Seyed Sadegh Salehi Amiri.
> Department of  Mathematics, Islamic Azad University, Babol Branch, Iran.
> salehisss at yahoo.com
> salehisss at baboliau.ac.ir
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