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
