On the Normal Structure of NonCommutative Association Schemes of Rank 6

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.