Indeed, for PSL(2, "p") acts non-trivially on "p" points if and only if "p" = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple – further, it does not act non-trivially on "fewer" than "p" points. This was first observed by Évariste Galois in his last letter to Chevalier, 1832. This can be analyzed as follows; note that for 2 and 3 the action is not faithful (it is a non-trivial quotient, and the PSL group is not simple), while for 5, 7, and 11 the action is faithful (as the group is simple and the action is non-trivial), and yields an embedding into "Sp". In all but the last case, PSL(2, 11), it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on "p" points: Further, "L"2(7) and "L"2(11) have two "inequivalent" actions on "p" points; geometrically this is realized by the action on a biplane, which has "p" points and "p" blocks – the action on the points and the action on the blocks are both actions on "p" points, but not conjugate (they have different point stabilizers); they are instead related by an outer automorphism of the group. More recently, these last three exceptional actions have been interpreted as an example of the ADE classification: these actions correspond to products (as sets, not as groups) of the groups as "A"4 × Z/5Z, "S"4 × Z/7Z, and "A"5 × Z/11Z, where the groups "A"4, "S"4 and "A"5 are the isometry groups of the Platonic solids, and correspond to "E"6, "E"7, and "E"8 under the McKay correspondence. These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces), respectively: the compound of five tetrahedra inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary Fano plane) inside the Klein quartic (genus 3), and the order 3 biplane (Paley biplane) inside the buckyball surface (genus 70).