Hyperovals on H(3,q2) left invariant by a group of order 6(q+1)3

Three new infinite families of hyperovals on the generalized quadrangle H(3,q2) (q=ph, p a prime) of sizes 6(q+1), 12(q+1)2 (if q>7) and 6(q+1)2 (if p>3) are constructed. Furthermore they turn out to be invariant under the action of a linear collineation group of order 6(q+1)3 that fixes no point or line in a secant plane of H(3,q2). In particular the hyperovals of size 6(q+1)2 are transitive.