In mathematics, the generalized symmetric group is the wreath product
of the cyclic group of order m and the symmetric group of order n. There is a natural representation of elements of
as generalized permutation matrices, where the nonzero entries are m-th roots of unity:
The representation theory has been studied since (Osima 1954); see references in (Can 1996).
As with the symmetric group, the representations can be constructed in terms of Specht modules; see (Can 1996).
factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to
(concretely, by taking the product of all the
values), while the sign map on the symmetric group yields the
These are independent, and generate the group, hence are the abelianization.
The second homology group (in classical terms, the Schur multiplier) is given by (Davies & Morris 1974): Note that it depends on n and the parity of m:
which are the Schur multipliers of the symmetric group and signed symmetric group.