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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 that this does indeed form such a Steiner system, that is, that any $5$ elements are contained in a unique one of the above blocks. Here are my thoughts so far: note that the 6 elements in our starter block are just the non zero squares and $\infty$, so that it is stabilized in $PSL_2(11)$ by at multiplication by squares. As $PSL_2(11)$ has order $660$, this shows there are at most $660/5 = 132$ blocks. In fact the stabilizer of a block has size only $5$, which I can verify but in a slightly computational way. If one does that, it suffices to show no two distinct blocks intersect in $5$. Another helpful fact is that $PGL_2(11)$ acts $3$ transitively on the $12$ points of the projective line, and $PSL_2(11)$ can at least map any ordered $3$ to any other unordered $3$ (in $3$ ways), which seems helpful, but I can't quite show what I want from this. Wikipedia gives a reference to this statement, but it doesn't contain a proof.

I care about this result as I know a handful of constructions of $S(5,6,12)$ already, as well as its uniqueness, thus understanding result will further shed light on this Steiner system's symmetries, which form the sporadic group $M_{12}$.

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