Let Ω be a finite set of size n. A cyclic permutation on Ω is a permutation whose cycle decomposition is one cycle of length n. This paper classifies all finite doubly-transitive permutation groups which contain a cyclic permutation. The classification appears in Table 1.
We use (G, Ω) for a finite doubly-transitive permutation group G acting on a finite set Ω. For other notation and definitions see the self-contained article Cameron .
McSorley, John P. "Cyclic Permutations in Doubly-Transitive Groups." (Jan 1997).