The Group C_s (a)

Silviu Radu managed to do a complete analysis of the 18,345,885,696 different cubes of this subgroup which has the highest number of 20f*-maneuvers from all symmetric subgroups. Here the the table with the distances, the average distance is 17.87:

Up to M-symmetry there are 2,292,846,080 cubes which exactly have this symmetry and of these there are 23793 cubes which need 20 moves. All other cubes can be solved in 19 moves or less. 7391 out of these 23793 cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 15592 20f*-cubes, which are included in the file 20moves.zip. We display only some examples of this huge symmetry class here.

The next table gives the number of cubes up to M-symmetry which exactly have C_s (a)-symmetry. To compute this number we used the following identity, where C_s (a)Total is the table above and the right side hold the tables for the number of cubes mod M with exactly the symmetry of the subgroups of C_s (a), which are all known except for C_s (a) itself.

C_s (a)Total = 8 C_s (a) + 4 C_2h (a) + 4 C_2v (b) + 8 C_2v (a2) + 6 D_2h (face) + 2 D_2h (edge) +
4 D_2d (edge) + 2 C_4h + 4 C_4v + 3 D_4h + 2 T_h + O_h

The average maneuver length also is 17.87:

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© 2017  Herbert Kociemba