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===Group structure===
The set of ''n''×''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'',''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a [[normal subgroup]]. Indeed, the generalized permutation matrices are the [[normalizer]] of the diagonal matrices, meaning that the generalized permutation matrices are the ''largest'' subgroup of GL in which diagonal matrices are normal.
The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''<sup>×</sup> and ''S''<sub>''n''</sub>. Concretely, this means that it is the [[semidirect product]] of Δ(''n'', ''F'') by the [[symmetric group]] ''S''<sub>''n''</sub>:
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