Example of non-commutative association scheme

Question

I need an example of non-commutative association scheme of ordered 6. I tried to use the example in the book Handbook of Combinatorial Designs, Second Edition by Charles J. Colbourn‏،Jeffrey H. Dini but I could not reach the answer.

I took $G=\mathbb{Z_{6}}$ acts transitively on itself by $g\oplus x$ for all g and x in $\mathbb{Z_6}$. $G$ acts naturally on the set$ \mathbb{Z_6} \times \mathbb{Z_6}$ by $g(x_1,x_2)=(g\oplus x_1,g\oplus x_2)$ for all $g ,x_1$ , and $x_2$ in $\mathbb{Z_6}$. Letting $R_i=O((x,y))$ for all (x,y) in $\mathbb{ Z_6} \times \mathbb{Z_6}$ Where the O((x,y))is the orbit set of (x,y). I got 6 relations which make a commutative assocation scheme.

Answer 1

Everything I've found suggests that the notion of an association scheme is a generalization of a group.

That being the case, then the group of symmetries of an equilateral triangle is an example.

Answer 2

Here are all association schemes of order 6: http://math.shinshu-u.ac.jp/~hanaki/as/data/as06 Clearly a symmetric association scheme is also commutative. So you have to check just No.4 and 6 in the catalogue to figure out which one is non-commutative.