  The classification of flag-transitive generalised quadrangles is a
long-standing open problem at the interface of finite geometry and permutation
group theory. Given that all known flag-transitive generalised quadrangles are
also point-primitive (up to point-line duality), it is likewise natural to seek
a classification of the point-primitive examples. Working towards this aim, we
are led to investigate generalised quadrangles that admit a collineation group
$G$ preserving a Cartesian product decomposition of the set of points. It is
shown that, under a generic assumption on $G$, the number of factors of such a
Cartesian product can be at most four. This result is then used to treat
various types of primitive and quasiprimitive point actions. In particular, it
is shown that $G$ cannot have holomorph compound O'Nan-Scott type. Our
arguments also pose purely group-theoretic questions about conjugacy classes in
non-Abelian finite simple groups, and about fixities of primitive permutation
groups.
