or $n$ is odd.

2.  If $\gcd(p,n) = 2$ then $G$ has a GIM only if $q$ is even.

3.  If $\gcd(p,n) = 3$ then $G$ has a GIM if and only if $(r,p,q,n)$ is $$(3,3,3,3) \text{ or } (6,3,3,3) \text{ or } (6,6,3,3) \text{ or }(6,3,6,3).$$

4.  If $\gcd(p,n) = 4$ then $G$ has a GIM only if $r \equiv p \equiv q \equiv n \equiv 4 \modu 8).$

5.  If $\gcd(p,n) \geq 5$ then $G$ does not have a GIM.

In arriving at this result, we prove a useful criterion for determining conjugacy in $G(r,p,n)$ and give an explicit description of the automorphism group of $G(r,p,q,n)$; see Proposition \[grpnconjugacy\] and Theorem \[vN=N\] below. Parts (1) and (2) of this theorem are proved as Corollary \[part1-cor\] and Proposition \[part2-prop\], while parts (3)-(5) comprise Theorem \[main-thm\]. We note as a corollary that the theorem provides a complete classification when $q$ or $n$ is odd. This shows that projective reflection groups which are not self-dual may still possess GIMs.

Let $G=G(r,p,q,n)$ and assume $n \neq 2$ and $(r,p,q,n)$ is not one of the four exceptions $(3,3,3,3)$ or $(6,3,3,3)$ or $(6,6,3,3)$ or $(6,3,6,3)$. If $q$ or $n$ is odd, then $G$ has 