Isomorphisms Between Subiaco q –Clan Geometries S. E. Payne ∗ Tim Penttila Ivano Pinneri † Abstract For q =2 e , e ≥ 4, the Subiaco construction introduced in [2] provides one q–clan, one flock, and for e ≡ 2 (mod 4), one oval in PG(2,q). When e ≡ 2 (mod 4), there are two inequivalent ovals. The associated generalised quadrangle of order (q 2 ,q) has a complete automorphism group G of order 2e(q 2 − 1)q 5 . For each Subiaco oval O there is a group of collineations of PG(2,q) induced by a subgroup of G and stabilising O. When e ≡ 2 (mod 4), for both ovals the complete stabiliser is just that induced by a subgroup of G . 1 Introduction In [2] a new family of Subiaco q –clans, q =2 e , were introduced. Associated with a q –clan C is a generalised quadrangle GQ(C) of order (q 2 ,q ), subquadrangles of order q and their accompanying ovals in PG(2,q ), a flock F (C) of a quadratic cone in PG(3,q ), a line spread in PG(3,q ), and a whole variety of related translation planes. These various geometries derived from a Subiaco q –clan are all referred to as Subiaco geometries. In the present work we concentrate on the generalised * The first author enjoyed the warm hospitality of both the Combinatorial Computing Research Group of The University of Western Australia and C. M. O’Keefe of The University of Adelaide, as well as the financial support of the Australian Research Council. He also gratefully acknowledges the friendly atmosphere of the Department of Pure Mathematics and Computer Algebra of the Universiteit of Ghent and the support of the Belgian National Fund For Scientific Research during the evolution of this paper. † The third author acknowledges the support of a University of Western Australia Research Scholarship. Received by the editors May 1994 Communicated by J. Thas Bull. Belg. Math. Soc. 2 (1995), 197–222