Generalized symmetric group

Definition
The generalized symmetric group with parameters $$m$$ and $$n$$, denoted $$S(m,n)$$, is defined in the following equivalent ways:


 * 1) It is the defining ingredient::external wreath product of the defining ingredient::cyclic group $$\mathbb{Z}_m$$ of order $$m$$, and the defining ingredient::symmetric group $$S_n$$ (specifically, defining ingredient::symmetric group on finite set) of degree $$n$$, with the natural action of the latter on a set of size $$n$$.
 * 2) It is the defining ingredient::external semidirect product of the defining ingredient::homocyclic group $$(\mathbb{Z}_m)^n$$ and the symmetric group of degree $$n$$, where the latter has a natural action by coordinate permutations.
 * 3) It is the subgroup of the general linear group $$GL(n,\mathbb{C})$$ over the field of complex numbers comprising monomial matrices (i.e., matrices where every row has exactly one nonzero entry and every column has exactly one nonzero entry) where all the nonzero entries are $$m^{th}$$ roots of unity. Note that the group also has a faithful monomial representation of degree $$n$$ over any field where the polynomial $$x^m - 1$$ splits completely.
 * 4) It is the centralizer inside the symmetric group of degree $$mn$$ of a permutation that is a product of $$n$$ disjoint cycles of size $$m$$ each.

Two very special cases

 * In the case $$n = 1$$, we get the usual symmetric group (specifically, the symmetric group on finite set).
 * In the case $$n = 2$$, we get the signed symmetric group.

GAP implementation
These groups can be constructed in GAP using the WreathProduct, CyclicGroup and SymmetricGroup functions, as follows:

WreathProduct(CyclicGroup(m),SymmetricGroup(n))