Wolfgang Knapp writes:

Dear GAP forum,

for dealing with primitive permutation groups with the help of GAP it is
important to have an efficient way to implement in GAP the action of
degree m^k of the natural wreath product
WreathProduct(Sym(m),Sym(k),IdendityMapping(Sym(k))),
usually called "product action" (I think "power action" is a more appropriate
name). Is it possible to avoid complicated and complex computations in
defining this action?

I would be grateful for any useful advice, W. Knapp

I don't know whether there is a standard GAP function to do this, but
there does happen to be one in the share-package "smash".

It is called WreathPower(G,P) - G can be either a permutation or a
matrix group, and P must be a permutation group.

The source is in the file main-gap-directory/pkg/smash/code/c_wreath.g
if you want to look at it.

Here is an example:

gap> RequirePackage("smash");
gap> G:=SymmetricGroup(3);
Group( (1,3), (2,3) )
gap> P:=SymmetricGroup(4);
Group( (1,4), (2,4), (3,4) )
gap> W:=WreathPower(G,P);
Group( ( 1, 3)( 4, 6)( 7, 9)(10,12)(13,15)(16,18)(19,21)(22,24)(25,27)(28,30)
(31,33)(34,36)(37,39)(40,42)(43,45)(46,48)(49,51)(52,54)(55,57)(58,60)(61,63)
(64,66)(67,69)(70,72)(73,75)(76,78)(79,81), ( 2, 3)( 5, 6)( 8, 9)(11,12)
(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)
(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)(65,66)(68,69)(71,72)(74,75)(77,78)
(80,81), ( 2,28)( 3,55)( 5,31)( 6,58)( 8,34)( 9,61)(11,37)(12,64)(14,40)
(15,67)(17,43)(18,70)(20,46)(21,73)(23,49)(24,76)(26,52)(27,79)(30,56)(33,59)
(36,62)(39,65)(42,68)(45,71)(48,74)(51,77)(54,80), ( 4,28)( 5,29)( 6,30)
( 7,55)( 8,56)( 9,57)(13,37)(14,38)(15,39)(16,64)(17,65)(18,66)(22,46)(23,47)
(24,48)(25,73)(26,74)(27,75)(34,58)(35,59)(36,60)(43,67)(44,68)(45,69)(52,76)
(53,77)(54,78), (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)
(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)
(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72) )
gap> Size(W);
31104

Derek Holt.