# The Group Cs (b)

Silviu Radu managed to do a complete analysis of the 424,673,280 different cubes of this subgroup.Here the the table with the distances, the average distance is 17.71:

 Distance Number Distance Number 0f 1 11f 6,478 1f 2 12f 25,130 2f 1 13f 114,700 3f 0 14f 546,558 4f 1 15f 3,102,247 5f 4 16f 19,703,503 6f 24 17f 110,503,448 7f 50 18f 252,385,203 8f 126 19f 38,279,025 9f 533 20f 4,290 10f 1,956

Up to M-symmetry there are 106,103,792 cubes which exactly have this symmetry and 948 of these cubes need 20 moves. All other cubes can be solved in 19 moves or less. 778 out of these 948 cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 863 20f*-cubes, which are included in the file 20moves.zip. We display only some nicer examples of this symmetry class here.

The next table gives the number of cubes up to M-symmetry which exactly have Cs (b)-symmetry. To compute this number we used the following identity, where Cs (b)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(b), which are all known except for C(b) itself.

C(b)Total = 4 C(b) + 2 C2h (b) + 2 C2v (b) + 4 C2v (a1) + 2 D2h (edge) + 2 D2d (face) + 2C4v + D4h + 4 C3v+ 2 D3d + Oh

The average maneuver length also is 17.71:

 Distance Number Distance Number 0f 0 11f 1,439 1f 0 12f 5,906 2f 0 13f 28,154 3f 0 14f 135,189 4f 0 15f 772,641 5f 0 16f 4,919,061 6f 3 17f 27,611,515 7f 9 18f 63,069,968 8f 20 19f 9,558,403 9f 117 20f 948 10f 419
 Name shortest maneuver with exactly this symmetry Generator U B2 U D B2 D' (6f*)
 Name Generator B2 R D R' F2 R2 B' R' B' L2 R B' U2 B2 U' R' (16f*)
 Name Generator D' B F' R D U' B2 R2 D2 U2 F' L R' D R2 U2 (16f*)
 Name Generator L' B L' B2 F R D' B U R' F' L U' F2 (14f*)
 Name Generator L B2 D L' F' R D R B2 F' L2 U B2 U R2 (15f*)
 Name Generator L F U' F U2 L' R2 B' U B L2 R2 U L2 D' U2 (16f*)
 Name Generator B' R' U R' U2 B' R D' L2 R D' F2 U L2 R2 U2 (16f*)
 Name Generator F' L' F2 L2 B2 D' U B L' R D L2 D U2 R (15f*)
 Name Generator U' L2 D L' R F' R2 F2 D' U L' B' F L2 R2 (15f*)
 Name Generator U B2 D2 F' R U B2 F2 D' L' F D2 F2 U' (14f*)
 Name Generator U' F2 U' B' F R F2 L2 D' U F L R' U2 (14f*)

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© 2017 Herbert Kociemba