The cubature formulas on a sphere invariant with respect to a dihedral group of rotations with inversion $D_{6h}$

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a dihedral group of rotations with inversion D6h has been veloped. This algorithm was applied to find parameters of all the best cubature formulas of this group of symmetry up to the $23$rd order of accuracy $n$. In the course of the study carried out, the exact values of parameters of the corresponding cubature formulas were found for $n\le11$, and the approximate ones were obtained by the numerical solution of systems of nonlinear algebraic equations by a Newton-type method for the other $n$. For the first time, the ways of obtaining the best cubature formulas for the sphere were systematically investigated for the case of the group which is not a subgroup of the groups of symmetry of the regular polyhedrons.