  The study of Fuglede's conjecture on the direct product of elementary abelian
groups was initiated by Iosevich et al. For the product of two elementary
abelian groups the conjecture holds. For $\mathbb{Z}_p^3$ the problem is still
open if $p\ge 11$. In connection we prove that Fuglede's conjecture holds on
$\mathbb{Z}_{p}^2\times \mathbb{Z}_q$ by developing a method based on ideas
from discrete geometry.
