I have an undirected graph where each node is labelled with an integer key and I'm asked to detect every simple 4-cycle, which can be seen as an empty square (i.e. the two opposite nodes of the cycle have not to be connected).
I know the labels of a generic square in the graph can be permuted in $4! = 24$ ways, but on his paper (pages 33-36) Cohen states that the symmetric group $S_4$ has $S_4 = 8$ elements, and this should allow me to focus only on $24/8 = 3$ distinct cases.
I don't understand this last point. Well, I know he probably means that the suqare has $8$ isometries (four reflections and four rotations), but I don't understand why this leads to the three cases in Figure 7 of the paper.