# The Group C2 (a)

Silviu Radu managed to do a complete analysis of the 15,288,238,080 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.49:

 Distance Number Distance Number 0f 1 11f 2,602,404 1f 6 12f 9,295,556 2f 15 13f 30,949,920 3f 72 14f 101,692,042 4f 343 15f 363,402,817 5f 1,060 16f 1,333,918,995 6f 4,570 17f 4,550,677,966 7f 17,960 18f 7,774,262,733 8f 60,136 19f 1,120,268,931 9f 226,199 20f 51,094 10f 805,260

Up to M-symmetry there are 1,910,682,624 cubes which exactly have this symmetry and 5763 of these cubes need 20 moves. All other cubes can be solved in 19 moves or less. 3165 out of these 5763 cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 4464 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 (a)-symmetry. To compute this number we used the following identity, where C2 (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(a), which are all known except for C(a) itself.

C2 (a)Total = 8 C2 (a) + 4 C2v (a2) + 4 C2h (a) + 4 S4 + 12 D2 (face) + 4 C4 + 4 D2 (edge) + 4 C2v (a1) + 6 D2h(face) + 2 D2d (edge) + 2D2h (edge) + 2C4h + 2C4v + 6 D2d (face) +
6D4 + 3D4h + 4 T + 2 Th + Oh

The average maneuver length also is 17.49:

 Distance Number Distance Number 0f 0 11f 323,655 1f 0 12f 1,158,307 2f 0 13f 3,863,416 3f 8 14f 12,702,003 4f 34 15f 45,411,240 5f 123 16f 166,705,822 6f 507 17f 568,759,399 7f 2,182 18f 971,648,345 8f 7,210 19f 139,966,952 9f 28,017 20f 5,763 10f 99,641
 Name shortest maneuver with exactly this symmetry Generator L R U2 (3f*)
 Name Generator B2 U' L R' B D' U L2 B2 R' B' F U' R2 (14f*)
 Name Generator D U' B F D' B2 F2 U B F D U' (12f*)
 Name Generator U2 B F' L2 R2 B F' U (8f*)
 Name Generator F2 L' R D2 F' D2 U2 R2 D U' F2 L U2 B F' R2 (16f*)
 Name Generator U' L' R D2 U2 L R' D' (8f*)
 Name Generator U' B2 F2 D' (4f*)
 Name Generator D U2 B2 F2 U' (5f*)

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© 2017 Herbert Kociemba