Conjugacy class of transpositions is the unique smallest conjugacy class of involutions

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Statement

Let n \ge 7. The conjugacy class of Transposition (?)s in the Symmetric group (?) on a set of size n, is the Conjugacy class (?) of smallest size among all conjugacy classes of involutions (elements of order two). Moreover, there is no other conjugacy class of the same size.

This fails for n = 6, where the conjugacy class of triple transpositions has the same size. For n = 3 it is the only conjugacy class of involutions, while for n = 4,5, the conjugacy class of double transpositions is strictly smaller in size.