or (ii) $n=2$ and either $p$ or $r/p$ is odd; this is the statement of [@Ma Theorem 5.2].

Deducing this theorem from [@Ma Theorem 5.2] is straightforward, given our next main result. Let $r,n$ be positive integers and let $p,p',q,q'$ be positive divisors of $r$ such that $pq=p'q'$ divides $rn$. The following result simplifies and extends [@Ca1 Theorem 4.4]; its proof occupies Sections \[iso1-sect\], \[iso2-sect\] and \[iso3-sect\].

\[thm2\] The projective reflection groups $G(r,p,q,n)$ and $G(r,p',q',n)$ are isomorphic if and only if either (i) $ \gcd(p,n) = \gcd(p',n)$ and $\gcd(q,n) = \gcd(q',n)$ or (ii) $n=2$ and the numbers $p+p'$ and $q+q'$ and $\frac{r}{pq}$ are all odd integers.

As a corollary, we can say precisely when the group $G(r,p,q,n)$ is “self-dual” as in Theorem \[thm1\].

\[cor2\] The projective reflection group $G=G(r,p,q,n)$ is isomorphic to its dual $G^*=G(r,q,p,n)$ if and only if either (i) $\gcd(p,n) = \gcd(q,n)$ or (ii) $n=2$ and $\frac{r}{pq}$ is an odd integer.

On seeing Theorem \[thm1\] one naturally asks whether for arbitrary projective reflection groups the property of having a GIM is equivalent to self-duality. Theorem \[thm2\] allows us to attack this question directly; its answer turns out to be false, and the rest of our results are devoted 