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

A ternary relational structure X is an ordered pair (V,E) where V is a set and E a set of ordered 3-tuples whose coordinates are chosen from V (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 G if [inline image], the automorphism group of X, contains the left regular representation of G. We prove that two Cayley ternary relational structures of [inline image], p>=11 a prime, are isomorphic if and only if they are isomorphic by a group automorphism of [inline image]. This result then implies that any two Cayley digraphs of [inline image] are isomorphic if and only if they are isomorphic by a group automorphism of [inline image], p>=11 a prime.