Using Burnsides Theorem
How many distinguishable regular octagons can be painted if each edge of a octagon is painted one of eleven colours and different edges can be the same colour?
the symmetric group of a regular octagon is the dihedral group of order 16, which is generated by two permutations and
that is for each suppose is the number of cycles in the complete decomposition of then Burnside's lemma says
that the number of coloring for a regular octagon using 11 colors is as an example, there are 8 cycles (of length 1) in the complete
decomposition of the identity element of and so more examples: and have one cycle and 5 cycles respectively and thus etc.