Given $ I\subseteq S_n$ subset of permutations that only have $\frac{(n^2-n)}{2}$ inversions. How can I describe the order of the generating subgroup $\langle I \cup (\,1 2)\ \rangle$?
I know that $I$ has n elements, but think that the permutation $(\,1 2)\in I $ when $ n\geq 2$ so the order of $\langle I \cup (\,1 2)\ \rangle$ is n?