My question is this:
Coronene consists of 24 carbon atoms and 12 hydrogen atoms arranged as illustrated in the link below. If we allow some or all of the hydrogen atoms to be replaced by chlorine, how many different isomers can be formed? To answer this, let S be the set of the 12 carbon atoms which have a non-carbon attachment. Identify all symmetries of this figure, free to move in 3 dimensions, as permutations of S. Write the pattern inventory for all 2-colorings of S, and determine the total number of 2-colorings.
But I can't figure out where to begin finding the symmetries on this thing. Do we go through the edges and the vertices? It seems that going through each vertex goes through the vertex on the opposite side. I'd really appreciate some help finding these symmetries; once I find those, I think that I could easily find the permutations of them, the cycle index, pattern inventory, number of 2-colorings, etc.