The Group S4

Silviu Radu did most of the analysis of the 442368 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.31:

Distance
Number
Distance
Number
0f
1
11f
522
1f
0
12f
1,224
2f
3
13f
3,990
3f
0
14f
8,076
4f
1
15f
20,147
5f
0
16f
52,237
6f
10
17f
125,622
7f
36
18f
176,669
8f
94
19f
52,945
9f
127
20f
428
10f
236

Up to M-symmetry there are109376 cubes which exactly have this symmetry. All but 82 cubes up to M-symmetry can be solved with less than 20 moves. 68 out of these 82 cubes with 20 moves have antisymmetry. 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 S4-symmetry, the average maneuver length is 17.32 now:

Distance
Number
Distance
Number
0f
0
11f
110
1f
0
12f
256
2f
0
13f
937
3f
0
14f
1,944
4f
0
15f
4,965
5f
0
16f
12,900
6f
0
17f
31,218
7f
7
18f
43,854
8f
16
19f
13,014
9f
25
20f
82
10f
48

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file S4.zip. The 75 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

Name
shortest maneuver with exactly this symmetry
Generator
U R2 L2 U2 R2 L2 D (7f*)
Name
Generator
F L B' D2 R D U' B' D2 L F' R' (12f*)
Name
Generator
D B U2 B' D' F D2 B F' U2 B' U F D2 F' U' (16f*)
Name
Generator
B' D U' L' B L R F' L2 U' F2 L F2 R2 U2 L F2 U (18f*)
Name
Generator
F L F2 D B' F R' F U' F D2 U F2 D' U F' R' (17f*)
Name
Generator
L' U' L' R' D' B' L R' B U L R D R (14f*)
Name
Generator
F L B' D2 R D U' B' D2 L F' R' (12f*)
Name
Generator
U2 F L B' D2 R D U' B' D2 L F' R' U2 (14f*)

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© 2017  Herbert Kociemba