  In this paper we characterize the orbit codes as geometrically uniform codes.
This characterization is based on the description of all isometries over a
projective geometry. In addition, the Abelian orbit codes are defined and a new
construction of Abelian non-cyclic orbit codes is presented. In order to
analyze their structures, the concept of geometrically uniform partitions have
to be reinterpreted. As a consequence, a substantial reduction in the number of
computations needed to obtain the minimum subspace distance of these codes is
achieved and established. An application of orbit codes to multishot subspace
codes obtained according to a multi-level construction is provided.
