# The Group Ci

With the help of GAP and methods similar to the methods used in Cube Explorer to find optimal maneuvers, Silviu Radu managed in a very ingenious way to do a complete analysis of the 45,864,714,240 different cubes . Here the the table with the distances, the average distance is 17.86:

 Distance Number Distance Number 0f 1 11f 788,992 1f 0 12f 3,315,172 2f 9 13f 13,445,694 3f 0 14f 54,961,138 4f 63 15f 271,122,047 5f 120 16f 1,585,605,201 6f 694 17f 9,134,397,172 7f 2,562 18f 27,801,243,853 8f 11,338 19f 6,999,321,491 9f 44,853 20f 259,100 10f 194,740

Up to M-symmetry there are 1,910,931,706 cubes which exactly have the symmetry of this subgroup. All of them can be solved within 20 moves, 10540 up to M-symmetry need exactly 20 moves. Up to M-symmetry and M-antisymmetry it are 7188 20f*-cubes, which are included in the file 20moves.zip. Below are some nice examples for cubes, which exactly have Ci symmetry.

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

CiTotal = 24 Ci+ 12 C2h (b) + 12 C2h (a) + 6 D2h(face) + 6 D2h (edge) + 6 C4h +
3 D4h + 8 S6 + 4 D3d + 2 Th + Oh

The average maneuver length also is 17.86:

 Distance Number Distance Number 0f 0 11f 32,617 1f 0 12f 137,692 2f 0 13f 559,379 3f 0 14f 2,288,348 4f 1 15f 11,293,497 5f 4 16f 66,058,983 6f 22 17f 380,581,022 7f 102 18f 1,158,344,598 8f 440 19f 291,614,579 9f 1,846 20f 10,540 10f 8,036
 Name shortest maneuver with exactly this symmetry Generator U D' R L' (4f*)
 Name Generator R2 B L' R F D2 B R U F' R2 U' B L B L2 U' (17f*)
 Name Generator R2 U2 L U2 B' D2 U' L2 U' F' D2 L B' U2 B2 R2 (16f*)
 Name Generator F D F' U B L R2 D F' R2 F D' R U' F2 (15f*)
 Name Generator L' R U L U2 B L U B R' D L' U2 F' L2 D' U' (17f*)
 Name Generator B F L' D' U B' L R' U B' F R U2 (13f*)
 Name Generator L' R B' R2 B2 D' U R D2 U2 L B F U (14f*)
 Name Generator F L2 B' F2 U L R' F' U' R' F L2 R2 B' R' U' (16f*)
 Name Generator D2 U F2 D2 F' D2 L D2 F U2 R2 B2 U' B2 R' U2 (16f*)

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