The Group C₃

Silviu Radu did most of the analysis of the 3,779,136 different cubes of this subgroup. Here the the table with the distances, the average distance is 18.13:

Distance	Number	Distance	Number
0f	1	11f	26
1f	0	12f	162
2f	0	13f	288
3f	0	14f	1,711
4f	0	15f	8,924
5f	0	16f	67,176
6f	1	17f	481,063
7f	0	18f	2,083,458
8f	4	19f	1,133,447
9f	18	20f	2,829
10f	28

Up to M-symmetry there are 942716 cubes, which exactly have this symmetry. There are 659 20f*-cubes up to M-symmetry and 576 20f*-cubes up to M-symmetry and M-antisymmetry which exactly have C₃-symmetry.

The next table gives the number of cubes up to M-symmetry which exactly have C₃-symmetry, the average maneuver length still is 18.13:

Distance	Number	Distance	Number
0f	0	11f	5
1f	0	12f	32
2f	0	13f	71
3f	0	14f	418
4f	0	15f	2,216
5f	0	16f	16,705
6f	0	17f	120,050
7f	0	18f	520,076
8f	0	19f	282,475
9f	3	20f	659
10f	6

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file C3.zip (filesize 10.7 MB !!!). The 576 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

Below are some nice examples of cubes with exactly C₃-symmetry.

Name	shortest maneuver with exactly this symmetry
Generator	L' R U2 R2 D2 F2 L R D2 (9f*)

Name
Generator	U B' F D' F2 D B F' R' F2 R U' (12f*)

Name
Generator	D B' L2 F L F' L2 B L' D' (10f*)

Name
Generator	U R' F2 R F D' F2 D F' U' (10f*)

Name
Generator	U B L' D2 R F' D B' D2 F L' B L R' D' U' (16f*)

Name
Generator	R' B R F2 L F2 R' F D2 R' D' U' B R2 F' U (16f*)

Name
Generator	U2 B' L B' U R' U R U2 B2 L' U2 (12f*)

Name
Generator	D' R' B' F L2 B' L' B2 L' F' R D' (12f*)

Name
Generator	F U' R2 D L' B' F D U L' D2 L' R2 F' U' (15f*)

Name
Generator	D' B2 U R D B' L' B' L' D F U L2 D' (14f*)

Name
Generator	F2 D L D L B2 U R B D' U2 R U2 B L2 F U' (17f*)

Name
Generator	B' D' U L' R U' F D2 F' U F D2 B' U (14f*)

Name
Generator	D B U2 B' L D' U B2 D' U B' L' U2 L D (15f*)

Name
Generator	F D' U2 R B' F D' U' L' R B L R2 U' (14f*)

Name
Generator	F' D' R U2 L' B' F U R2 D' R U (12f*)

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The Group C3

The Group C₃