A pseudo-hyperoval of a projective space, q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that is flag-transitive and isomorphic to, where is either the regular hyperoval of PG(2, 4) or the Lunelli-Sce hyperoval of PG(2, 16).
Bamberg, J., GLASBY, S., Popiel, T., & Praeger, C. (2016). Generalized Quadrangles and Transitive Pseudo-Hyperovals. Journal of Combinatorial Designs, 24(4), 151-164. https://doi.org/10.1002/jcd.21411