A combinatorial approach to transitive extensions of generously unitransitive permutation groups

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H,Ω) on the κ-subsets of X, k=2,3,4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer-generated data for the particular series of Hamming schemes H(m,3), m≥2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya.