Ok, so a graph can represent the symmetries of a group in a total or a partial way, no problems tend to show up as long as there's support. The support for the structure of the graph, in the first rectangular 2x3 section, is either the direct product of two well-known groups of co-prime order, or its what you get when you rotate part of a Rubik's cube. It sounds kinda dumb, it's just a puzzle. Yes, but the graph's complexity increases dramatically even with low numbers in the vertex elements (m,n). The simplest part of the graph is the upper section mentioned, where the rotation group of a solid cube gets partitioned by--a partition function (?) This pops out when you restrict rotations in the physical model to 180°. You can keep it in one copy of \( S_4 \) by restricting again, to a pair of adjacent faces, and their opposite faces (hemispheres, I should say). So each of three copies of \( S_4 \) is wrapped this way, into a set of "face moves" around the same centre (acting, so to speak, on the sides of a box, not the top or bottom). This gives you the partition, and a way to weight the vertex subset along the boundary with numbers of elements in equivalence classes. The 2x2x2 puzzle has I think about 3.7 million permutations, and that's only the rotation subgroup. The graph is the structure of a table or matrix of numbers of these elements, the coset space with a partial order. K?