To start with, my dihedral is a bit specific, here is a picture

I need to find amount of ways to color faces ( there are 8 ) into 3 colours.

I have already something in my mind because of help Marko Riepel, here is my previous question, where he provided solution for a bit other solid, and one more, where solid is almost like mine, just its lower and upper parts are not truncated . So, I am not sure I have properly found rotations to make cycles index. And not sure about reflections, because in my task there is no word about .

I have these: 1) 2 rotations by 120 degrees, which change only the vericles in the middle triangle + identity e( nothing changes) 2) 3 rotations by 180 degrees passing through medians of triangles ( so 2 triangles vericles exchange and also small triangles exchange). So, there are 6, is it all? I misunderstand a bit which moves with solid are allowed and which arent.

Also, am I write, that the degree of cycle a depends of elements which werent changed? Because I dont truly understand the form of writing cycle index in the examples I linked to, I mean, as I found out, we work with group of permulations, and the amount of cycles in each permulations = degree for N in Burnsides lemma? For example, identity e permulation (1)(2)(3)(4)(5)(6)(7)(8) has 8 cycles so the first addend will be N^8?

Sorry guys, I feel really stupid, I spent whole day reading my institute lectures, googling example of use of lemma, searching in russian forums, foreign forums and still I cant solve this fucking task...


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