Generalized symmetric group

In mathematics, the generalized symmetric group is the wreath product formula_1 of the cyclic group of order "m" and the symmetric group of order "n".
Representation theory.
There is a natural representation of elements of formula_7 as generalized permutation matrices, where the nonzero entries are "m"-th roots of unity: formula_8
The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see .
Homology.
The first group homology group (concretely, the abelianization) is formula_9 (for "m" odd this is isomorphic to formula_10): the formula_11 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 formula_11 (concretely, by taking the product of all the formula_11 values), while the sign map on the symmetric group yields the formula_14 These are independent, and generate the group, hence are the abelianization.
The second homology group (in classical terms, the Schur multiplier) is given by :
Note that it depends on "n" and the parity of "m:" formula_17 and formula_18 which are the Schur multipliers of the symmetric group and signed symmetric group.