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This paper contains the classification of the orbits of elements of thetensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes{\mathbb{F}}^r$, $r\geq 1$, under the action of two natural groups, forall finite; real; and algebraically closed fields. For each of the orbits wedetermine: a canonical form; the tensor rank; the rank distribution of thecontraction spaces; and a geometric description. The proof is based on thestudy of the contraction spaces in${\mathrm{PG}}({\mathbb{F}}^2\otimes{\mathbb{F}}^3)$ and is geometric innature. Although the main focus is on finite fields, the techniques are mostlyfield independent.

Classification of subspaces in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3$ and orbits in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r$