What is the smallest $k$ such that we have $p^{(k)}=I d$ for all permutations of fixed length $n$ (we denote $p$ for the function corresponding to the permutation).
If for any permutation of length $n$ we have $p^{(k)}=I d$ then for any length $1 \leq i \leq n$ of a cycle of a permutation of length $n$ the number $k$ must be a multiple of $i$.

Indeed, all numbers $i \in\{1,2, \ldots, n\}$ can be the length of a cycle of a permutation of length $n$. Indeed, it suffices to take the permutation $(2,3, \ldots, i-1, i, 1, i+1, i+2, \ldots, n)$ which has the cycle $(1,2,3, \ldots, i)$ which is of length $i$.

This number is therefore $\operatorname{LCM}(1,2, \ldots, n)$. If we denote $p_{i}$ the $i^{\text {th }}$ prime number and $\alpha_{i}$ the greatest integer such that $p_{i}^{\alpha_{i}} \geq n$, we have that $\operatorname{LCM}(1,2, \ldots, n)=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}} . \cdots . p_{i}^{\alpha_{i}} . \cdots$.