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In this paper, we obtain classification results for higher-dimensionalanalogues of classical association schemes called association schemes ontriples (ASTs). We present an algorithm that enumerates all ASTs on a fixednumber of vertices whose nontrivial relations are invariant under the action ofsome group. Applying this algorithm to three, four, and five vertices alongwith appropriate group actions yields the unique AST over three vertices, theunique symmetric ASTs over four or five vertices, the unique AST over fourvertices with two nontrivial relations, and the unique nontrivial circulant ASTover five vertices.

Association schemes on triples over few vertices