TITLE: Can all the sporadic groups be expressed as permutation groups based on a single big cycle?
QUESTION [8 upvotes]: Working on M11, I came up with that it can be generated using the following permutations:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[[2, 0, 1, 7], [3, 4, 5, 6]]
[[4, 0, 6, 7], [2, 3, 1, 5]]
[[0, 7], [4, 6], [3, 5], [1, 2]]
The summary of that is that there's one big permutation which cycles everything, and a few more which allow GF(8) on the first 8 elements.
Do all the other sporadic groups have generators with a single cycle of everything and a few which work on the minimum number of other elements possible? If so, what are they?

REPLY [20 votes]: If a permutation group $G$ on a set of size $n$ contains an $n$-cycle then the subgroup $C$ generated by this $n$-cycle is transitive. Thus we have a factorisation $G=CG_\alpha$. Moreover, $C\cap G_\alpha=1$. All factorisations of the sporadic simple groups were determined by me in my paper 'Factorisations of sporadic simple groups' in the Journal of Algebra in 2006. This shows that the 11-cycle in the action of $M_{11}$ on 11 points and the 23-cycle in the action of $M_{23}$ on 23 points are the only examples.