Generalized quadrangles with a point-primitive and line-primitive automorphism group with socle $\PSp_4(q)$

Abstract

Let $\mc{S}$ be a finite thick generalized quadrangle, and let $G\leq \Aut(\mc{S})$ act primitively on both points and lines. Building on the almost simple reduction for point-primitive and line-primitive actions, we study the case where the socle of $G$ is the projective symplectic group $\PSp_4(q)$ with $q\ge 3$. We show that, up to duality, either $\mc{S}\cong W(3,q)$ for $q\ge 3$, or $q=3$ and $\mc{S}\cong H(3,4)$.