The complex reflection groups were fully classified by Shephard & Todd (1954) : each complex reflection group is isomorphic to a product of irreducible complex reflection groups, and every irreducible either belongs to an infinite family G ( m , p , n ) (where m, p, and n are positive integers such that p divides m) or is one of 34 other (so-called "exceptional") examples. The group G ( m , 1 , n ) is the generalized symmetric group : algebraically, it is the wreath product ( Z / m Z ) ≀ ≀ S n of the cyclic group Z / m Z with the symmetric group S n . Concretely, the elements of the group may be represented by monomial matrices (matrices having one nonzero entry in every row and column) whose nonzero entries are all m th roots of unity. The groups G ( m , p , n ) are subgroups of G ( m , 1 , n ) , and in particular the group G ( m , m , n ) consists of those matrices in which the product of the nonzero entries is equal to 1.