$p$ and $p'$ with equal multiplicity, and so we have in fact that $
{\mathrm{GCD}}(\eta p,p') = {\mathrm{GCD}}(\eta p',p) = {\mathrm{GCD}}(p,p')$.

The next proposition implies the converse of Theorem \[isomo\], provided $n\neq 2$.

Assume $n\neq 2$ and let $G={G(r,p,q,n)}$.

1.  The center of $G$ has order $\frac{r}{pq} \cdot {\mathrm{GCD}}(p,n)$.

2.  The abelianization $ G/[G,G]$ of $G$ has order $ \frac{2r }{pq} \cdot {\mathrm{GCD}}(q,n)$.

One can easily check that, since $n\neq 2$, the center of $G$ is given by the set of its scalar elements (i.e. of the form $c^i$). The number of scalar elements in $G$ is $\frac{1}{q}$ times the number of scalar elements in $G(r,p,n)$, which is $\frac{r}{p} \cdot {\mathrm{GCD}}(p,n)$ by [@Ma Corollary 4.1].

To prove (2), it suffices to count the linear characters of $G$ since these are equal in number to the order of $G/[G,G]$. By [@Ca1 §6], the linear characters of $G(r,n)$ are parametrized by $r$-tuples of partitions $(\lambda_0,\ldots,\lambda_{r-1})$ where all partitions $\lambda_i$ are empty except one which can be either $(n)$ or $(1^n)$. The linear representations of $G(r,1,q,n)$ are parametrized by these $r$-tuples of partitions where, if the only non-empty partition appears in a position $i$, then $ni\equiv 0\modu q)$ (i.e. $(\lambda_0,\ldots,\lambda_{r-1})\in {\mathrm{Fer}}(r,q,1,n)$ in the 