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In the projective space $\mathrm{PG}(3,q)$, we consider orbits of lines underthe stabilizer group of the twisted cubic. In the literature, lines of$\mathrm{PG}(3,q)$ are partitioned into classes, each of which is a union ofline orbits. We propose an approach to obtain orbits of the class named$\mathcal{O}_6$, whose complete classification is an open problem. For all evenand odd $q$ we describe a family of orbits of $\mathcal{O}_6$ and theirstabilizer groups. The orbits of this family include an essential part of all$\mathcal{O}_6$ orbits.

Orbits of the class $\mathcal{O}_6$ of lines external with respect to the twisted cubic in $\mathrm{PG}(3,q)$