Canonical forms of 2×3×32×3×3 tensors over the real field, algebraically closed fields, and finite fields

We classify the orbits of elements of the tensor product spaces F2⊗F3⊗F3F2⊗F3⊗F3 for all finite, real, and algebraically closed fields under the action of two natural groups. The result can also be interpreted as the classification of the (GL(F2)×GL(F3)×GL(F3))(GL(F2)×GL(F3)×GL(F3))-orbits in the 17-dimensional projective space of 2×3×32×3×3 tensors, which is the natural home of the Segre product of a projective line and two projective planes. The proof is geometric in nature, relies on properties of the Segre embedding, and uses the terminology of projective spaces.