Examples of rank 3 product action transitive decompositions

A transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This characterisation showed that every such decomposition arose from a 2-transitive decomposition of Km via one of two general constructions. Here we use results of Sibley to give an explicit classification of those which arise from 2-transitive edge-decompositions of Km.