Generalized symmetric group
Introduction In mathematics, the '''generalized symmetric group''' is the wreath product  of the cyclic group of order ''m'' and the symmetric group of order ''n''.
Examples * For  the generalized symmetric group is exactly the ordinary symmetric group: * For  one can consider the cyclic group of order 2 as positives and negatives () and identify the generalized symmetric group  with the signed symmetric group.
Representation theory 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 ; 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  (for ''m'' odd this is isomorphic to ): the  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 :::Note that it depends on ''n'' and the parity of ''m:''  and  which are the Schur multipliers of the symmetric group and signed symmetric group.
References * * *