Spherical 7-design in the 2n-dimensional Euclidean space

We consider a finite subgroup Θ_n of the group O(N) of orthogonal matrices, where N=2ⁿ, n=1, 2, ... . This group was defined in [4] and we use it to construct spherical designs in the 2ⁿ-dimensional Euclidean space R^N. We prove that representations ρ₁, ρ₂ and ρ₃ of the group Θ_n on the spaces of harmonic polynomials of degrees 1, 2 and 3 respectively are irreducible. This together with the earlier results [1, 3] imply that the orbit Θ_n,2x of any initial point x on the unit sphere S_N-1 is a 7-design in the Euclidean space of dimension 2ⁿ