Topological representations of polyhedral isomerizations

Now consider the symmetry point group G (or, more precisely, the framework group [70]) of the above ML coordination compound or -vertex cluster compound. This group has G operations of which R are proper rotations so that G / R = 2 if the compound is achiral and G / R = 1 if the compound is chiral (i.e., has no improper rotations). The distinct permutations of the n sites in the coordination compound or cluster are divided into / / right cosets [71] which represent the permutational isomers since the permutations corresponding to the i proper rotations of a given isomer do not change the isomer but merely rotate it in space. This leads naturally to the concept of isomer count, I, namely 

Development of top-reps for isomerizations of polyhedra having more than six vertices is complicated by large isomer counts. Thus for the most symmetrical and chemically significant seven-vertex deltahedra the isomer counts are 7 / D5 = 5040/10 = 504 for the pentagonal bipyramid and 7 / C3 = 5040/3 = 1680. In this connection a baby monster graph with 1680 vertices has actually been described in the chemical literature as a model for degenerate rearrangements in the seven-vertex system [73]. 

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