  In the paper Graphical complexity of products of permutation groups, M.
Grech, A. Jez, A. Kisielewicz have proved that the direct product of
automorphism groups of edge-colored graphs is itself the automorphism groups of
an edge-colored graph. In this paper, we study the direct product of two
permutation groups such that at least one of them fails to be the automorphism
group of an edge-colored graph. We find necessary and sufficient conditions for
the direct product to be the automorphism group of an edge-colored graph. The
same problem is solved for the edge-colored digraphs.
