Fissioned triangular schemes via sharply 3-transitive groups

In [D. de Caen, E.R. van Dam, Fissioned triangular schemes via the cross-ratio, European J. Combin. 22 (2001) 297–301], de Caen and van Dam constructed a fission scheme FT(q+1) of the triangular scheme on PG(1,q). This fission scheme comes from the naturally induced action of PGL(2,q) on the 2-element subsets of PG(1,q). The group PGL(2,q) is one of two infinite families of finite sharply 3-transitive groups. The other such family M(q) is a “twisted” version of PGL(2,q), where q is an even power of an odd prime. The group PSL(2,q) is the intersection of PGL(2,q) and Mq(q). In this paper, we investigate the association schemes coming from the actions of PSL(2,q), Mq(q) and PML(2,q), respectively. Through the conic model introduced in [H.D.L. Hollmann, Q. Xiang, Association schemes from the actions of PGL(2,q) fixing a nonsingular conic, J. Algebraic Combin. 24 (2006) 157–193], we introduce an embedding of PML(2,q) into PML(3,q). For each of the three groups mentioned above, this embedding produces two more isomorphic association schemes: one on hyperbolic lines and the other on hyperbolic points (via an orthogonal parity) in a 3-dimensional orthogonal geometry. This embedding enables us to treat these three isomorphic association schemes simultaneously.