or $p'$. By Theorem \[isomo\], if $q'$ is the unique odd parameter then $G'\cong G(r,pq,1,2)$, and if $p'$ is the unique odd parameter then $G'\cong G(r,1,pq,2)$, and in either case $G \cong G(r,\frac{pq}{2},2,2)$. It thus suffices to show that if $p$ and $\frac{r}{p}$ are even then $G(r,2p,1,2)\not\cong G(r,p,2,2)$ and $G(r,1,2p,2)\not\cong G(r,p,2,2)$. With these hypotheses on $p$ and $\frac{r}{p}$, let $$\begin{aligned} A &= \{ g^{r/p} : g \in G(r,2p,1,2)\}, \\ B &= \{ g^{r/p} : g \in G(r,1,2p,2)\},\\ C &= \{g^{r/p} : g \in G(r,p,2,2)\}.
  \end{aligned}$$ As in the proof of Lemma \[npnq\], it is not difficult to check that $A$ consists of the distinct elements $(1;\frac{ir}{p},-\frac{ir}{p}) \in G(r,2p,2)$ for $i \in [p]$. On the other hand, one finds similarly that $B$ consists of the distinct images in $G(r,1,2p,2)$ of the elements $(1;0,\frac{ir}{p}) \in G(r,2)$ for $i \in [p]$. Finally, $C$ consists of the distinct images in $G(r,p,2,2)$ of the elements $(1;\frac{ir}{p},-\frac{ir}{p}) \in G(r,p,2)$ for $i \in [\frac{p}{2}]$. Thus $|A| = |B| = p$ and $|C| = \frac{p}{2}$, which establishes the desired non-isomorphisms.

We now examine a particular class of groups $G=G(r,p,2)$ where we can explicitly describe an isomorphism $\phi:G\rightarrow G^*$.

\[coprime\] If $p$ or $\frac{r}{p}$ is odd then $G(r,p,1,2) \cong 