[GAP Forum] regular subgroups of the Symmetric Group S_24 isomorphic to S_4.

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear Alejandra,
On Sun, Sep 04, 2016 at 08:16:31PM -0300, Alejandra Alderete wrote:
> 
> I need to work with groups of hight order and my computer isn't  potency o
> capacity , I don't  know. I am sending the algorithm. I need to find the
> regular subgroups of the Symmetric Group  S_24  isomorphic to S_4.
> 
> gap> s4 := SymmetricGroup (4);
> gap> s24 := SymmetricGroup (24);
> gap> homo := AllHomomorphisms (s4, s24);;
> gap> inj :=Filtered (homo, function (v) return IsInjective (v)= true ; end);
> gap> img := List (inj, x-> Image (x, s4));
> gap> reg := Filtered (img, function (v) return IsRegular(v)= true ; end);

This would be a hopelessly long list. There are 24!/576 such subgroups in
S_24.
On the other hand constructing one copy is very easy, just let S_4 act
on itself.
gap> S4:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> Action(S4,Elements(S4),OnRight);
Group([
(1,10,17,19)(2,9,18,20)(3,12,14,21)(4,11,13,22)(5,7,16,23)(6,8,15,24),
(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,15)(14,16)(17,18)
(19,21)(20,22)(23,24) ])

Hope this helps,
Dmitrii.


> 
> 
> 
> best regards
> 
> Alejandra