Permutation Power

Computer Science Level 4

Let $\sigma$ be a permutation of $\{1, 2, 3, \ldots, 100\}$ . Then there exists a smallest positive integer $f(\sigma)$ such that $\sigma^{f(\sigma)} = \text{id}$ , where $\text{id}$ is the identity permutation.

What is the largest value of $f(\sigma)$ over all $\sigma?$

Note: $\sigma^k$ is $\sigma$ applied $k$ times in succession.