The search for the best cubature formulae invariant under the octahedral group of rotations with inversion for a sphere

The definition of the best cubature formula invariant under the octahedral group of rotations with inversion for a sphere is given. The process of searching for the best cubature formulae of the given symmetry type is described. The table which contains the main characteristics of all the best today cubature formulae of the octahedral group of rotations with inversion up to the 53rd algebraic order of accuracy is given. The weights and the coordinates of the new cubature formulae of the 21st, 25th, 27th, 31st and 33rd orders of accuracy are given to 16 significant digits.