A generalization of a construction due to Van Nypelseer (Q2568868)

Starting with a (thick) rank two geometry \(\mathcal S\) the author describes a construction of an incidence structure \(\Gamma(\mathcal S)\) of rank 3 as follows: a flag \(C=(p,L)\) is incident with a line \(L'\), if and only if \(L \cap L' = p\); \(C=(p,L)\) is incident with a point \(p'\) if and only if \(p' \in L\) and \(p' \neq p\); a point \(p\) and a line \(L\) are incident if and only if \(p \not\in L\). This construction was first used by \textit{L. Van Nypelseer} and \textit{C. Lefèvre-Percsy} in the case of projective planes [Discrete Math. 84, No. 2, 161--167 (1990; Zbl 0712.51006)]. It turns out that if \(\Gamma (\mathcal S)\) is a firm, residually connected, (IP)\(_2\) and flag-transitive geometry, then \(\mathcal S\) is either a linear space or a \((4,3,4)\)-gon. The author provides a complete classification of all Steiner systems \({\mathcal S} = S(t,k,v)\) with \(t > 2\) for which \(\Gamma (\mathcal S)\) is a firm, residually connected, (IP)\(_2\) and flag-transitive geometry. When applied to the Steiner systems associated to the Mathieu groups \(M_{24}\), the construction produces a new firm, residually connected rank six geometries on which \(M_{24}\) acts flag-transitively.