Further Results on Orbits and Incidence Matrices for the Class O6 of Lines External to the Twisted Cubic in PG(3,q)

In the literature, lines of the projective space PG(3, q) are partitioned into classes, each of which is a union of line orbits under the stabilizer group of the twisted cubic. The least studied class is named O6. This class contains lines external to the twisted cubic which are not its chords or axes and do not lie in any of its osculating planes. For even and odd q, we propose a new family of orbits of O6 and investigate in detail their stabilizer groups and the corresponding submatrices of the point-line and plane-line incidence matrices. To obtain these submatrices, we explored the number of solutions of cubic and quartic equations connected with intersections of lines (including the tangents to the twisted cubic), points, and planes in PG(3, q).