# How do I represent monomial reflection groups?

Let m and n be positive integers, let $C_m$ be the cyclic group of order m and B = $C_m$ × . . . × $C_m$ be the direct product of n copies of $C_m$. For each divisor p of m define the group A(m, p, n) by

A(m, p, n) = {($θ_1$, $θ_2$, . . . , $θ_n$) ∈ B | ($θ_1$$θ_2$ . . . $θ_n$)$^{m/p}$ = 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 $m^nn!/p$. It is well known that these groups can be realized as finite subgroups of $GL_n(C)$, 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.

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