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Wikipedia gives a beautiful construction of the $S(5,6,12)$ Steiner system; take as blocks the subset of the projective line over the field with $11$ elements consisting of $\{\infty,1,3,4,5,9\}$ and all its images under the natural action of $PSL_2(11)$. I am hoping to find a slick verification...

In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered. Sometimes the notation M9, M10, M20 and M21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct...