Geometries for the group PSL(3,4)

We classify all firm, residually connected coset geometries, on which the group PSL(3,4) acts as a flag-transitive automorphism group fulfilling the residually weakly primitive condition: the stabilizer of any flag F acts primitively on the elements of some type in the residue ΓF. We demand also that every residue of rank two satisfies the intersection property. We give geometric constructions for all geometries obtained.