Double transposition in symmetric group:S4

We consider the group $$G$$ defined as symmetric group:S4, i.e., the symmetric group of degree four, which for convenience we take to be the symmetric group acting on the set $$\{ 1,2,3,4\}$$.

We are interested in the conjugacy class of double transpositions for this group, i.e., permutations whose cycle decomposition comprises two disjoint 2-cycles. These form a single conjugacy class (see cycle type determines conjugacy class) and also form a single orbit under the action of the automorphism group of $$S_4$$.

The complete list of elements is:

$$\{ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$