Non-degenerate 2 × k × (k + 1) hypermatrices

We show that if F is a topological field, then there is a transitive, free and continuous action of a natural quotient of GLk(F)×GLk+1(F) on the set Mk(F) of 2×k×(k+1) hypermatrices over F with non-zero hyperdeterminant. We use this action to study the homotopy type of Mk(C) and Mk(R) and count elements of Mk(Fq) (generalizing an unpublished result of Lewis and Sam).