Incidence matrices for the class O6 of lines external to the twisted cubic in PG(3, q)

We consider the structures of the plane-line and point-line inci-dence matrices of the projective space PG(3 , q) connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all q , even and odd, we determine the incidence matrices connected with a family of orbits of the class named O6. This class contains lines external to the twisted cubic. The consid-ered family includes an essential part of all O6 orbits, whose complete classification is an open problem.