Abstract
There is a regular graph with 12 vertices and valency 6 which has three mutually orthogonal 1-factorizations. Any pair of these can be interpreted as a Howell design or semi-Latin square. The automorphism group of the graph is A 5 × Z 2; it preserves the above three 1-factorizations as a set, interchanging two of them. This can be interpreted as a semi-Latin square isomorphic to its transpose with a unique Latin square orthogonal to it.
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