The Group C_s (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 C_s (b)-symmetry. To compute this number we used the following identity, where C_s (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_s(b), which are all known except for C_s(b) itself.

C_s(b)Total = 4 C_s(b) + 2 C_2h(b) + 2 C_2v(b) + 4 C_2v(a1) + 2 D_2h(edge) + 2 D_2d(face) + 2C_4v + D_4h + 4 C_3v+ 2 D_3d + O_h

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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The Group Cs (b)

The Group C_s (b)