A combinatorial approach to doubly transitive permutation groups

Let G be a doubly but not triply transitive group on a set X. We give an algorithm to construct the orbits of G acting on X x X x X by combining those of its stabilizer H of a point of X if the group H is given first, we compute the orbits of its transitive extension G, if it exists. We apply our algorithm to G = PSL(m, q) and Sp(2m, 2), m >= 3, successfully. We go forward to compute the transitive extension of G itself. In our construction we use a superscheme defined by the orbits of H on X x X x X and do not use group elements. (c) 2007 Elsevier B.V. All rights reserved.