Four infinite families of non-Schurian association schemes of order 2p^2
Association schemes are one of the traditional areas of investigation in algebraic graph theory.
Catalogues of all association schemes are available from the website of Hanaki and Miyamoto. It is known that all association schemes of order up to 14 are Schurian, that is they are coming from suitable transitive permutation groups in the standard manner. First examples of non-Schurian association schemes exist on 15, 16 and 18 vertices. In particular, there are just two classes of non-Schurian association schemes of order 18.
Starting from successful computer free interpretations of these two examples, and using extensive computer algebra experiments in conjunction with further reasoning, we were able to observe the existence of at least four infinite families of non-Schurian association schemes of order 2p^2 (for p prime, p>3).
This is joint work with M. Klin.