# The Group C2 (b)

Silviu Radu managed to do a complete analysis of the 2,548,039,680 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.82:

 Distance Number Distance Number 0f 1 11f 28,146 1f 0 12f 119,054 2f 3 13f 473,702 3f 0 14f 2,168,270 4f 1 15f 12,440,909 5f 0 16f 84,769,773 6f 44 17f 548,185,762 7f 180 18f 1,583,371,387 8f 460 19f 316,458,701 9f 1,587 20f 13,628 10f 8,072

Up to M-symmetry there are 636,937,008 cubes which exactly have this symmetry and 3297 of these cubes need 20 moves. All other cubes can be solved in 19 moves or less. 2153 out of these 3297cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 2725 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 C2 (b)-symmetry. To compute this number we used the following identity, where C2 (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.

C2 (b)Total = 4 C2 (b) + 4 D2 (edge) + 2 C2h (b) + 2C2v (b) + 2 D2d (edge) + 2 D2h (edge) + 2 D4 + 4 D3 + D4h + 2 D3d + Oh

The average maneuver length also is 17.82:

 Distance Number Distance Number 0f 0 11f 6,713 1f 0 12f 29,113 2f 0 13f 117,581 3f 0 14f 540,239 4f 0 15f 3,107,127 5f 0 16f 21,185,220 6f 2 17f 137,030,323 7f 33 18f 395,812,880 8f 78 19f 79,102,176 9f 365 20f 3,297 10f 1,861
 Name shortest maneuver with exactly this symmetry Generator U R2 D' U' R2 U' (6f*)
 Name Generator R2 B2 D' B' D B' L2 F U' F' L2 R2 (12f*)
 Name Generator U' F' L' R D R2 B2 L2 U L' R F' U (13f*)
 Name Generator F' D2 L2 D' U L' F L' F' L B2 U' B2 D' F (15f*)
 Name Generator F2 D B' F' R' F2 D2 L' B2 D U F' L' R' U F2 (16f*)
 Name Generator R2 F2 U' L2 R' D' F L2 F' D L' F2 L' R' (14f*)
 Name Generator B' D' R' D' U B F L2 R2 F' U R (12f*)
 Name Generator F L R' B R B' D' B2 F U' L R' D' F U (15f*)

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© 2017 Herbert Kociemba