Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ s.t. } ({\bf P}^m)_{ii} = 1 \text{ for any } i$$
will $\bf P$ need to be permutation similar to any representation for a generating element of a cyclic group of order $n$?