Permutation Power

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.

×