On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp6(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ⋀3V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ⋀3V.