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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