[GAP Forum] Cartesian Group Direct Product

[Alexander Hulpke]

Dear Paul Hjelmsted, Dear GAP-Forum,

> The problem with this is that it merely seems to shuffle between the  
> non-Cartesian
> form of the permutations, that is, for example, sending (123) to  
> (567) (Second Embedding) or merely leaving at at (123) (First  
> Embedding) but I may be doing something wrong.
>
> I am not getting anything Cartesian-wise. Perhaps I must leave D4  
> (actually called Dihedral(8)),
> as a pc-group and not a perm group?
>
> After I get this right, I need to generate the CycleIndex, and then  
> expand it in a manner you indicated, to get the full Polya  
> Polynomial, whose coefficients will be useful to me (especially to  
> find how many octads there are under D8 X S3 (Dihedral(16) X  
> Symmetric(3)) and other issues
>
> I need these generators for D4 X S3:
>
> (0,3,6,9)(1,4,7,10)(2,5,8,11)
> (0,4,8)(1,5,9)(2,6,10)(3,7,11)
> (1,7)(3,9)(5,11)
> (1,11)(2,10)(3,9)(4,8)(5,7)
>
> Or something with the same meaning

OK. You want a different representation for the direct product. GAP  
gives you by default the intransitive action (which has smaller  
degree), you would prefer the transitive product action.

The easiest way to construct this group is to let the intransitive  
direct product act on the cartesian product of the domains:

gap> d4:=DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> s3:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> d:=DirectProduct(d4,s3);
Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])
gap> cart:=Cartesian([1..4],[5..7]);
[ [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 5 ],
   [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ] ]
gap> hom:=ActionHomomorphism(d,cart,OnTuples,"surjective");
<action epimorphism>
gap> prod:=Image(hom);
Group([ (1,4,7,10)(2,5,8,11)(3,6,9,12), (4,10)(5,11)(6,12),
   (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2)(4,5)(7,8)(10,11) ])


Up to labelling (which is due to the arrangement of the pairs in  
`cart') these are the generators you listed. You can use the map `hom'  
to go back to the intransitive direct product and use its  
decomposition functions.


Best wishes,

     Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke