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A cubic C in PG(1, q) is the zero locus of a homogenous polynomial f(X0, X1) of degree 3 in Fq[X0, X1]. The cubic forms on PG(1, q) form a four-dimensional vector space W, and subspaces of the projective space PG(W) are called linear systems of cubics. One-dimensional linear systems are called pencils. In this talk, we mention combinatorial invariants of the equivalence classes of pencils of cubics on PG(1, q), for q odd and q not divisible by 3. These equivalence classes are considered as orbits of lines in PG(3, q), under the action of the subgroup G ∼= PGL(2, q) of PGL(4, q) which preserves the twisted cubic C in PG(3, q). In particular we determine the point orbit distributions and plane orbit distributions of all G-orbits of lines which, are contained in an osculating plane of C, have non-empty intersection with C, or are imaginary chords or axes.