A monomial matrix with entries in a field$F$ is an $n \times n$matrix with exactly one nonzero entry in each row and exactly one nonzero entry in each column. Under matrix multiplication, monomial matrices form a group; this group is isomorphic to a wreath product

$S_n \ltimes (F^\ast)^n$

with respect to the canonical action of the permutation group$S_n$ on the $n^{th}$ power $(F^\ast)^n$ of the group of nonzero scalars.