Cubature Formulas on the Sphere that are Invariant under Dihedral Rotation Groups

A process of searching on the sphere for the best (in a sense) cubature formulas that are invariant under the transformations of different dihedral rotation groups is described. The parameters of these new cubature formulas of the 6th, 10th, and 12th order of accuracy are given to 16 significant digits. A table which contains the main characteristics of all the best to date cubature formulas of the dihedral rotation group up to the 29th order of accuracy is given.