3-cycle in symmetric group:S3

We consider the group $$G$$ defined as symmetric group:S3, i.e., the symmetric group of degree three, which for convenience we take to be the symmetric group acting on the set $$\{ 1, 2, 3 \}$$.

We are interested in the conjugacy class of 3-cycles in this group, i.e., permutations that move all elements and have order three. The 3-cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of $$S_3$$.

The full list of elements in the conjugacy class is:

$$\{ (1,2,3), (1,3,2) \}$$