Characterization of p-valenced schemes of order p2q

In [P.-H. Zieschang, Association schemes in which the thin residue is a finite cyclic group, J. Algebra 324 (2010) 3572–3578], it was shown that association schemes are schurian if their thin residues are finite groups and have a distributive normal subgroup lattice. However, in the case that their thin residues are elementary abelian groups, they do not have a distributive normal subgroup lattice. Actually, they are not necessarily schurian due to the non-schurian quasi-thin scheme of order 28. In this paper, we investigate association schemes such that their thin residues are elementary abelian groups of order p2. It is shown that these schemes are schurian if the number of elements with valency p is 2p and the factor scheme over the thin residue is a cyclic group of prime order. We also construct non-schurian p-schemes of order p3.