where N is a cyclic group whose order is a factor of d and G' is a subgroup of PGL(2, C), i.e., a cyclic group [Z.sub.m], a dihedral group D2m, the tetrahedral group [A.sub.4], the octahedral group S4 or the
icosahedral group [A.sub.5].
The lepton families correspond to the 3-D finite binary rotational groups called the binary tetrahedral group 2T, the binary octahedral group 2O, and the binary icosahedral group 2I, also labelled as [3, 3, 2], [4, 3, 2], and [5, 3, 2], respectively, in Table 1.
The 120 operations of the binary icosahedral group 2I are represented by 120 unit quaternions, and 2I contains almost all the rotation operations needed for the 7 fermion family groups.