Regular hamiltonian embeddings of Kn,nKn,n and regular triangular embeddings of Kn,n,nKn,n,n

We give a group-theoretic proof of the following fact, proved initially by methods of topological design theory: Up to isomorphism, the number of regular hamiltonian embeddings of Kn,nKn,n is 22 or 11, depending on whether nn is a multiple of 88 or not. We also show that for each nn there is, up to isomorphism, a unique regular triangular embedding of Kn,n,nKn,n,n.