Consider the the set $$X = \prod_{1 \leq k \leq n-2} \binom{ \bf{n}}{k} $$ where $\binom{ \bf{n}}{k}$ denotes the set of subsets with $k$ elements of the set ${\bf n} = \{1, \cdots , n\}$.

For each $i \in {\bf n}$ and each permutation $g \in S_{{\bf n} \backslash \{i\}}$, we define a permutation $\tilde{g_k}$ of $\binom{\bf n}{k}$ as follows: if the $k$-subset $Y$ of ${\bf n}$ contains $i$ then $\tilde{g_k}$ fixes $Y$, otherwise $\tilde{g_k}$ sends $Y \subset {\bf n} \setminus \{i\}$ to $g(Y)$. The cartesian product of the $\tilde{g_k}$ taken together define a permutation $\tilde{g}$ of $X$. For each $i$, define a subgroup $G_i$ of the permutation group $S_X$ as follows:

$$G_i = \{ \tilde{g} : g \in S_{{\bf n} \backslash \{i\}} \};$$

note that $G_i \cong S_{n-1}$. Let $G = \langle G_i : i \in {\bf n}\rangle$ be the subgroup of $S_X$ generated by all these subgroups.

Can you describe $G$ and its action on $X$? Thank you in advance.