$r$ and $n$. We write $ {\mathbb{Z}}_r$ to denote the cyclic group of order $r$; for convenience we view this as the set $\{0,1,\dots,r-1\}$, with addition computed modulo $r$. Likewise we write $ S_n$ to denote the [symmetric group]{} of permutations of the set $[n]$.

Recall the definition of the group $G(r,n)$ from Example \[exa2\] in the introduction. This group is isomorphic to the semidirect product of $({\mathbb{Z}}_r)^n$ by $S_n$ with respect to the natural action of the symmetric group and we frequently employ the following notation to refer to its elements.

Given $\pi \in S_n$ and $x=(x_1,x_2,\dots,x_n) \in ({\mathbb{Z}}_r)^n$, let $$(\pi,x) \in G(r,n)$$ denote the matrix whose $i$th column has $(\zeta_r)^{x_i}$ in row $\pi(i)$ and zeros in all other rows, where $\zeta_r {\stackrel{\rm def}{=}}\exp\(\frac{2\pi  \sqrt{-1}}{r}\) $ is a fixed primitive $r$th root of unity.

When describing elements of $({\mathbb{Z}}_r)^n$, we often write $e_1,e_2\dots ,e_n$ for the standard basis of the free ${\mathbb{Z}}$-module $({\mathbb{Z}}_r)^n$, so that we may then express the element $x=(x_1,x_2,\dots,x_n) \in ({\mathbb{Z}}_r)^n$ as $x = \sum_{i=1}^n x_i e_i$.

The product of two elements $(\pi,x), (\sigma,y) \in G(r,n)$ is described as follows. The symmetric group $S_n$ acts on $({\mathbb{Z}}_r)^n$ by permuting coordinates; denote this action by letting $$\pi(x) 