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To be precise, the generalized permutation matrices are a (faithful) [[linear representation]] of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.
===Subgroups===
* The subgroup where all entries are 1 is exactly the [[permutation matrices]], which is isomorphic to the symmetric group.
* The subgroup where all entries are ±1 is the [[signed permutation matrices]], which is the [[hyperoctahedral group]].
* The subgroup where the entries are ''m''th [[roots of unity]] <math>\mu_m</math> is isomorphic to a [[generalized symmetric group]].
== Properties ==
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