Bill's post on nets reminds me of Jean Taylor's work Sci. Am. 1976 July on foam and bubbles. She has ten trivalent graphs on a sphere. Each has edges made of great arcs. They meet at 2.pi/3. Now it is "well-known" that the Cayley graph of the modular group (a free product <x.y> of a cyclic groups C2=<x> & C3=<y>), is a free trivalent tree with oriented nodes. (Each encoding for the triangle of the C3 action.) Any finite connected trivalent graph is the Schreier coset graph of the modular group on some subgroup. Each such graph yields permutations for the images of the two generators of the free product PSL(2,Z), generating a factor group of the modular group on cosets. We have (as Bill points out) for subgroups, the platonics, Gamma(N), N=2,3,4,5 yielding the regular representations of PSL(2,N), and the great circle itself {how to interpret this one?} and the "others": For the "other" graphs I find the disjoint cycle shapes (= cusp widths of the stabilizer of a point) to be of index: Index shape xy-image order of perm gp. comments 18: 44433 648 solvable 30: 5544444 2**3.3.5**3 solvable 36: 55553333 2**23.3**4.5.7 A9 composition factor 42: 555555444 2**10.3**9.5.7 A7 composition factor 48: 5555555544 2**14.3.5 ?solvable It would be fun if some minimality can be deduced from these groups alone! John