Non-commutative association schemes of rank 6 with affine subschemes

When association schemes are viewed as a generalization of groups, it becomes natural to seek non-commutative examples. As with groups, non-commutative association schemes must have at least six elements, but unlike in group theory, there are numerous examples with exactly six elements. One method to try to classify such schemes is to attempt to construct extensions of schemes of rank 3, starting with those schemes of rank 3 which correspond to self-complementary strongly regular graphs with a vertex-transitive automorphism group. Recent work of Klin, Kriger, and Woldar provides new constructions for such graphs. In this paper, we investigate the possibility of constructing new non-commutative schemes with six elements from these graphs.