notation of [@Ca1 §6]). Therefore the number of linear characters of $G(r,1,q,n)$ is $\frac{2r}{q}\cdot {\mathrm{GCD}}(q,n)$. One can likewise check that, since $n\neq 2$, each linear character of $G$ is given by the common restriction of exactly $p$ distinct linear characters of $G(r,1,q,n)$. Thus the number of linear characters of $G$ is $\frac{1}{p}$ times the number of linear characters of $G(r,1,q,n)$.

Combining the preceding theorem and proposition gives this corollary, which forms one half of Theorem \[thm2\] in the introduction.

\[cor3.5\] Assume $n\neq 2$ and $pq=p'q'$. Then $G(r,p,q,n)\cong G(r,p',q',n)$ if and only if ${\mathrm{GCD}}(p,n) = {\mathrm{GCD}}(p',n)$ and ${\mathrm{GCD}}(q,n) = {\mathrm{GCD}}(q',n)$.

Isomorphism classes in rank two {#iso2-sect}
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In this section we fix $n=2$, and assume that $p$, $p'$, $q$, $q'$ divide $r$ and $pq=p'q'$ divides $2r$. We now determine when the two groups $G=G(r,p,q,2)$ and $G'=G(r,p',q',2)$ are isomorphic.

In referring to elements of these groups, it is convenient to abbreviate our notation by writing $(\pi;a,b)$ for the element otherwise denoted $(\pi,(a,b)) \in G(r,p,q,2)$. We thus view $G(r,p,q,2)$ as the set of triples $(\pi;a,b) \in S_2 \times {\mathbb{Z}}_r \times {\mathbb{Z}}_r$ with $a+b$ divisible by $p$, where $(\pi;a,b) = (\pi';a',b')$ if and only if $\pi=\pi'$ and $a-a'\equiv b-b' \equiv k\frac{r}{q} \modu r)$ for 