How to actually construct the $S(3,6,22)$ Steiner system? The question is in the title. I see writte about these special Steiner systems quite frequently, but no explicit description how to construct them. In my case, I am especially interested in the system $S(3,6,22)$. How many sets does it even contain? Is it maybe infeasible to construct? * * * **My guess** Take the Mathieu group $M_{22}$ as a permutation group acting on $\\{1,2,...,22\\}$. I then take the orbit of an arbitrary $6$-element set. My hope would be that this gives me the desired Steiner system, but I have no reason to believe this.

One way would be to start with $S(5,8,24)$. Then fix a pair of elements. Throw away the octads not containing both of those, and then discard that pair from all the octads.

Of course, this assumes that you know how to construct $S(5,8,24)$. There are many relatively straight forward ways of doing that. One involves the extended binary Golay code. Another way that I happen to remember is to do it as three carefully chosen orbits of $C_{23}\rtimes C_{11}$. The details are given in this old answer of mine.

Your idea works otherwise but you are not allowed to start from an arbitrary set of six. Pick those six from an octad $B$ of $S(5,8,24)$. Then select two elements $u,v\in B$. Use the Mathieu group $M_{22}$ _that is exactly the stabilizer of both $u$ and $v$ in $M_{24}$._ Then apply that copy of $M_{22}$ to the sextet $B\setminus\\{u,v\\}$.

* * *

In other words, the choice of the sextet must match with the choice of a conjugate of $M_{22}$ within $S_{22}$.