  In this paper, we obtain classification results for higher-dimensional
analogues of classical association schemes called association schemes on
triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed
number of vertices whose nontrivial relations are invariant under the action of
some group. Applying this algorithm to three, four, and five vertices along
with appropriate group actions yields the unique AST over three vertices, the
unique symmetric ASTs over four or five vertices, the unique AST over four
vertices with two nontrivial relations, and the unique nontrivial circulant AST
over five vertices.
