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I am struggling to get my head fully around the conjugacy classes of D5.

Everywhere I have looked seems to say that there are 4 irreducible representations of D5 which implies that there are 4 conjugacy classes. However, when examining the symmetry of the pentagon I am only able to see 3 symmetries, namely the identity, reflections through an axis from a vertex to the mid-point of the opposite side and a rotation of 2*pi/5. In terms of permutations of a pentagon with vertexes labelled 1,2,3,4,5 clockwise, this would be (identity), (23)(45) and (12345).

http://mathworld.wolfram.com/DihedralGroupD5.html" [Broken] says that there are 4 conjugacy classes, but I cannot see what the extra one must be. Any light shed on this would be a great help for me as I do not have a huge amount of in depth knowledge about group theory but I have a basic understanding of other groups, but cannot figure this one out.

Thanks.

Edit * I have been trying to put further thought into it and one possible reason I have thought of is if the reflections through different axes are defined as a rotation and then just a reflection. This isn't how I have done things for a triangle or a square though, but it would, I think, give an extra set of actions that would have a different order to just a reflection, or just a rotation where the order of a reflection would be 2 and the order of a rotation would be 5 but a rotation and then a flip would have an order of 10 to get back to the identity. Does this sound feasible or am I barking up the wrong tree?

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# Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

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