: (\pi,x) \mapsto (\pi,-x)$.

Isomorphism classes of $G(r,p,q,n)$ {#iso1-sect}
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Let $r$ and $n$ be positive integers and let $p$, $p'$, $q$, $q'$ be positive integer divisors of $r$. Throughout, we assume $pq = p'q'$ and that this product divides $rn$, and we let $$G=G(r,p,q,n)
\qquad\text{and}\qquad
G'=G(r,p',q',n).$$ The main goal of this section is to determine a necessary and sufficient condition for $G$ and $G'$ to be isomorphic when $n \neq 2$. We will deal with the case $n=2$ in the next section. To begin, we recall from [@Ca1 Proposition 4.2] the following result.

\[suff\] If ${\mathrm{GCD}}(\frac{rn}{q},p')={\mathrm{GCD}}(\frac{rn}{q'},p)$ then for every $g\in G$ there exists a unique $g'\in G'$ such that $g$ and $g'$ have common representatives in $G(r,n)$, and in this case the map $g\mapsto g'$ determines an isomorphism $G\cong G'$.

Results in [@Ca1 §4] completely characterize when $G \cong G^*$ if $n\neq 2$ (where we define $G^* = G(r,q,p,n)$). Our strategy is to generalize the ideas in that work to the present context. Say that a prime integer $\pi$ appears in a number $k$ with multiplicity $e$ if $\pi^e$ divides $k$ and $\pi^{e+1}$ does not divide $k$. A prime is then *special* if it appears in $p$ and $p'$ 