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 thecompletedecomposition 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.