Twisted cubic and plane-line incidence matrix in $$\mathrm {PG}(3,q)$$

The point-plane, the point-line, and plane-line incidence matrices of $$\mathrm {PG}(3,q)$$ are interest in combinatorics, finite geometry, graph theory group theory. Some properties these their submatrices related with interplay orbits points, lines planes under action subgroups {PGL}(4,q)$$ . A remarkable particular case is subgroup $$G\cong \mathrm {PGL}(2,q)$$ , viewed as stabilizer twisted cubic $$\mathscr {C}$$ For this case, study point-plane matrix, initiated by D. Bartoli present authors, has attracted attention being to useful applications coding for construction multiple covering codes. In paper, we extend our investigation matrix apart from just one class line orbits, named $$\mathcal {O}_6$$ literature. all $$q\ge 2$$ each submatrix, numbers any plane through obtained. As a tool investigation, use incidences arising unions including such submatrix determine total number union average orbit union.