The Group C_i

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 C_i symmetry.

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

C_iTotal = 24 C_i+ 12 C_2h(b) + 12 C_2h(a) + 6 D_2h(face) + 6 D_2h(edge) + 6 C_4h +
3 D_4h + 8 S₆ + 4 D_3d + 2 T_h + O_h

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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The Group Ci

The Group C_i