The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p8p

A ternary relational structure XX is an ordered pair (V,E)(V,E) where VV is a set and EE a set of ordered 3-tuples whose coordinates are chosen from VV (so a ternary relational structure is a natural generalization of a 3-uniform hypergraph). A ternary relational structure is called a Cayley ternary relational structure of a group GG if Aut(X), the automorphism group of XX, contains the left regular representation of GG. We prove that two Cayley ternary relational structures of Z23×Zp, p≥11p≥11 a prime, are isomorphic if and only if they are isomorphic by a group automorphism of Z23×Zp. This result then implies that any two Cayley digraphs of Z23×Zp are isomorphic if and only if they are isomorphic by a group automorphism of Z23×Zp, p≥11p≥11 a prime.