Association schemes on triples (ASTs) are ternary analogues of classical
association schemes. Analogous to Schurian association schemes, ASTs arise from
the actions of two-transitive groups. In this paper, we obtain the sizes and
third valencies of the ASTs obtained from the two-transitive permutation groups
by determining the orbits of the groups' two-point stabilizers. Specifically,
we obtain these parameters for the ASTs obtained from the actions of Snβ and
Anβ, PGU(3,q), PSU(3,q), and Sp(2k,2), Sz(22k+1) and
Ree(32k+1), some subgroups of AΞL(k,n), some subgroups of PΞL(k,n), and the sporadic two-transitive groups. Further, we obtain the
intersection numbers for the ASTs obtained from these subgroups of PΞL(k,n) and AΞL(k,n), and the sporadic two-transitive groups. In
particular, the ASTs from these projective and sporadic groups are commutative.Comment: 20 pages, 5 table