Let m and n be positive integers, let Cm be the cyclic group of order m and B = Cm × . . . × Cm be the direct product of n copies of Cm. For each divisor p of m define the group A(m, p, n) by
A(m, p, n) = {(θ1, θ2, . . . , θn) ∈ B | (θ1θ2 . . . θn)m/p = 1}.
It follows that A(m, p, n) is a subgroup of index p in B and the sym- metric group Sym(n) acts naturally on A(m, p, n) by permuting the coordinates. G(m, p, n) is defined to be the semidirect product of A(m, p, n) by Sym(n). It follows that G(m, p, n) is a normal subgroup of index p in the wreath product Cm ≀ Sym(n) and thus has order mnn!/p. It is well known that these groups can be realized as finite subgroups of GLn(C), specifically as n × n matrices with exactly one non-zero entry, which is a complex m-th root of unity, in each row and column such that the product of the entries is a complex (m/p)th root of unity. Thus the groups G(m, p, n) are sometimes referred to as monomial reflection groups.