Hyperovals on left invariant by a group of order

Three new infinite families of hyperovals on the generalized quadrangle H ( 3 , q 2 ) ( q = p h , 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 , q 2 ) . In particular the hyperovals of size 6 ( q + 1 ) 2 are transitive.