The universal nonabelian representation of the Petersen type geometry related to J₄

Let G be a geometry with three points on each line. The universal representation group R(G) of 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 J₄ be the fourth sporadic simple group of Janko and let G(J₄) be the Petersen type geometry of J₄ whose points and lines are subgroups of order 2 and 2² in J₄ with normalizers containing Sylow 2-subgroups; the incidence relation is given by inclusion. We show that R(G(J₄)) is exactly J₄.