Some new non-schurian association schemes on 2p² points, p an odd prime, and related combinatorial structures

Let p be an odd prime. We provide a construction of four non-Schurian association schemes for every prime p ≥ 5 and two for p = 3. For p > 3 the construction is new, while for p = 3 it coincides with the non- Schurian schemes, obtained with the aid of a computer by A. Hanaki and I. Miyamoto. The discovered non-Schurian objects appear as algebraic mergings of the Schurian coherent configuration on 2p² points and rank 6p−2, which corresponds to the action of the Heisenberg group of order p³ on the set of points and lines of the classical biaffine plane. The results obtained are considered in a wider framework.