Let $p$ be a function associated with a permutation. We call the order of this permutation the smallest integer $k$ such that $p^{(k)}=I d$.

What is the largest order for a permutation of size 11?
To have $p^{(i)}(1)$, it is necessary (and sufficient) that $i$ be a multiple of the length of the cycle that contains 1. Since the sum of the lengths of the cycles of a permutation of length 11 is 11, the maximum order is the maximum LCM (Least Common Multiple) that can be achieved with numbers whose sum is 11. This maximum LCM is reached for 6 and 5. Therefore, the maximum order of a permutation of length 11 is $6 \times 5 = 30$.