Let $g_1, \ldots, g_n$ be permutations on $\Omega$ such that $$g_i^2 = 1 \quad \forall i \in [n]$$ $$g_i \not= g_j \quad \forall i, j \in [n] \text{ s.t. } i \not = j.$$

Let $G = \langle g_1, \ldots, g_n\rangle$ and suppose that $G$ is an abelian transitive group.

Question: Can $|G|$ be exponential in $n$ or is it always polynomial? More precisely, what is the largest $|G|$ possible?