On a finite group of matrices generating orbit codes on Euclidean sphere

We consider orbit codes on a sphere S^N-1 of radius 1 centered at the origin of N-dimensional Euclidean space R^N. These codes are constructed as follows. Let G be a finite group of orthogonal matrices and let x be a point on S^N-1. The orbit code is defined as 𝒦C(G,x)={gx;g∈G}, i.e. 𝒦(G,x) is an orbit of an initial point x under the action of the group G. Apparently Slepian (1968) was the first to define orbit codes. He named them group codes