Collineations of the Subiaco generalized quadrangles

Each generalized quadrangle (GQ) of order (q2, q) derived in the standard way from a conical flock via a q-clan with q = 2e has subquadrangles of order q associated with a family of q + 1 (not necessarily projectively equivalent) ovals in PG(2, q). A new family of these GQ is announced in [1] and named the Subiaco GQ. We begin a study of their collineation groups. When e is odd, e ≥ 5, the group is determined. In the standard notation for the GQ, the collineation group is transitive on the lines through the point (∞). As a corollary we have that up to the usual notions of equivalence, just one conical flock, one oval in PG(2, q), and one subquadrangle of order (q, q) arise.