Irreducible collineation groups fixing a hyperoval

Let G be an irreducible collineation group of a finite projective plane π of even order n≡0mod4. Our goal is to determine the structure of G under the hypothesis that G fixes a hyperoval Ω of π. We assume G≡0mod4. If G has no involutory elation, then G=O(G) S2 with a cyclic Sylow 2-subgroup S2 and G has a normal subgroup M of odd order such that a G/M has a minimal normal 3-subgroup. If the subgroup S generated by all involutory elations in G is non-trivial and Z(S) denotes its center, then either S Alt(6) and n=4, or S/Z(S) (C3×C3) C2, Z(S) is a (possibly trivial) 3-group and n≡1mod3. In the latter case there exists a G-invariant subplane in π such that the collineation group induced by G on is irreducible and fixes a hyperoval . Furthermore, the subgroup generated by all involutory elations in is a generalized Hessian group of order 18, that is and the configuration of the centers of the involutory elations in consists of the nine inflexions of an equianharmonic cubic of a subplane π0 of order 4. In particular, π0 is generated by the centers and the axes of all involutory elations in G, and hence it is the so-called Heringʹs minimal subplane of π with respect to G.