8
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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?

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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.

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This already fails for the second-smallest sporadic group $M_{12}$.

A simple subgroup $G$ of $S_N$ cannot contain an $n$-cycle with $n$ even (unless $|G|=2$...), because then $G \cap A_N$ would be an index-2 subgroup. In particular, if $G$ contains an $N$-cycle then $N$ is odd. But the largest odd number that occurs as the order of an element of $M_{12}$ is $11$, and $M_{12}$ is not a subgroup of $S_{11}$.

This might already be enough to prove that the only sporadic simple groups that do have such a representation are $M_{11}$ (as you found) and $M_{23}$.

[added later: in fact one doesn't even need the $G \cap A_N$ trick here: if a group $G$, simple or not, acts transitively on $N$ letters then the point stabilizer is an index-$N$ subgroup, and $M_{11}$ and $M_{23}$ are the only sporadic groups with an element of order large enough to be the index of a proper subgroup. For most sporadic $G$ it's not even close. For example, the largest proper subgroup of $J_1$ (third-smallest sporadic) has index $266$, but the maximal order of an element is $19$. Usually the discrepancy is even huger.

The $A_N$ trick is still useful for knowing a priori that (e.g.) $M_{12} \subset S_{12}$ cannot contain a 12-cycle if we know nothing about $M_{12}$ beyond its simplicity.]

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