The universal nonabelian representation of the Petersen type geometry related to \(J_ 4\) (Q1357800)

Let \({\mathcal G}\) be a geometry with three points on each line. The universal representation group \(R ({\mathcal G})\) of \({\mathcal G}\) is generated by a set of involutions, one for each point, subject to the relations that for every line the product of the three generators corresponding to the points on the line, is the identity. Let \(J_4\) be the fourth sporadic simple group of Janko and let \({\mathcal G} (J_4)\) be the Petersen type geometry of \(J_4\) whose points and lines are subgroups of order 2 and \(2^2\) in \(J_4\) with normalizers containing Sylow 2-subgroups; the incidence relation is given by inclusion. We show that \(R ({\mathcal G} (J_4))\) is exactly \(J_4\).