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×XX×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)G=PSL(m,q) and Sp(2m,2)Sp(2m,2), m⩾3m⩾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×XX×X×X and do not use group elements.