I'm experimenting (GAP3.4.4) with WreathProduct and SemidirectProduct and
seek to express the former in terms of the latter. A simple example
illustrates my question.

If S8 and S3 are respectively the symmetric groups on 8 and 3 letters, what
values of f and Z would equate WreathProduct(S8,S3,IdentityMapping(S3)) to
SemidirectProduct(DirectProduct(S8,S8,S8),f,Z), where f is a homomorphism
from DirectProduct(S8,S8,S8) into Aut(Z)?

Bruce