A \((c\cdot L^*)\)-geometry for the sporadic group \(J_2\)
Let \((J_2,{\mathcal G})\) be the natural permutation representation of the Hall-Janko sporadic simple group \(J_2\) on a strongly regular graph of degree 100 and valency 36. Every subgroup \(H\) of index 280 in \(J_2\) has an orbit of size 10 in \({\mathcal G}\). The authors now define the following rank 3 geometry \(\Gamma\): points are the vertices of \({\mathcal G}\); lines are the pairs of vertices contained in such an orbit of length 10; planes are these orbits of size 10 and incidence is natural. The authors show that the (flag-transitive) automorphism group of \(\Gamma\) is \(J_2.2\) (and also \(J_2\) acts flag-transitively, but no other subgroup), and that \(\Gamma\) has string diagram \(c\cdot L^*\), where the \(c\) stands for the complete graph on 10 vertices, and the \(L^*\) for the dual of the Hermitian unital on 28 points (related to \(U_3(3)\)). Some other information (geometric properties, but also the Boolean lattice of stabilizers) is provided.
scientific article; zbMATH DE number 1268415