Cubature formulas on the sphere that are invariant under the transformations of the dihedral groups of rotations with inversion

An algorithm for finding the best cubature formulas (in a sense) on the sphere that are invariant under the transformations of the dihedral groups of rotations with inversion is described. This algorithm is applied for finding parameters of all the best cubature formulas of this symmetry type up to the 35th order of accuracy.