# Conjugacy classes of rotational symmetry dodecahedron

My question is how to find all conjugacy classes of order 5 of the rotational symmetry of the dodecahedron. I know that I could find them by considering conjugacy classes of order 5 in A5, but I would like to find them by considering the dodecahedron.

I also know that the elements of order 5 are rotations through the middle of a side of the dodecahedron. My problem is I can't visualize which rotations I have to use to prove there are two conjugacy classes of elements of order 5.

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There are at least two conjugacy classes since the elements of order $5$ rotate through two different angles, $2\pi/5$ and $4\pi/5$.
As you say, the elements of order $5$ are rotations about the axes through the centres of the faces. We can consider each class as, say, the clockwise rotations through $2\pi/5$ and $4\pi/5$, respectively, about axes through the centre of each face, oriented towards that face.
A rotation through $\pi$ about an axis through the centre of an edge of a face swaps that face with the adjacent face with which it shares that edge. Any face can be moved to any other face by a series of swaps with adjacent faces.