Last line had some numbers wrong but is now fixed below. —Dan ----- Nice pictures in that paper! (I know nothing about group representations, and don't even know the definition of a wreath product.) Funny, just last night I was musing about how for any finite group G the fewest number N = N(G) of points on which it can act faithfully is an interesting invariant. Which is just finding G as a subgroup of the permutation group S_N. For instance, what is this number for the rotation groups of the regular polyhedra: T (order 12), O (order 24), I (order 60) ? Each group acts faithfully on the face centers, edge midpoints, and also the face centers of the dual polyhedron (for O and I). For the octahedron and icosahedron, and for the edges of the tetrahedron, antipodal points get rotated to antipodal points, so we can divide all these numbers by 2 except for the 4 face centers of the tetrahedron. Hence T permutes 4 or 3 = 6/2 points, O permutes 4 = 8/2 or 6 = 12/2 or 3 = 6/2 points, and I permutes 10 = 20/2 or 15 = 30/2 or 6 = 12/2 points. But wait! We can't represent T or O faithfully on 3 points! S_3 is only order 6, while these groups are bigger. Which of these 8 cases are faithful (represent all group elements as distinct permutations)? Can we do better than 4 for T (clearly not), 4 for O, and 6 for I ? -----