Flag-transitive hyperplane complements in classical generalized quadrangles

Let H be a geometric hyperplane of a classical finite generalized quadrangl e Q and let C = Q \ H be its complement in Q, viewed as a point-line geomet ry. We shall prove that C admits a flag-transitive automorphism group if an d only if H spans a hyperplane of the projective space in which Q is natura lly embedded (but with Q viewed as Q(4, q) when Q = W(q), q even). Furtherm ore, if Q is the dual of Ii(4, q(2)) and H, C are as above, then C is flag- transitive if and only if H = p(perpendicular to) for some point p of Q.