In the three-dimensional projective space PG(3,q) over the finite field F_q with q elements, we consider the normal rational curve known as a twisted cubic and the projectivity group G_q that fixes it. For q = 2, 3, 4, we solve the open problems of classifying the orbits of points, planes, and lines under G_q and of determining the corresponding incidence matrices between points, planes, and lines partitioned into these orbits.