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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The Group S4

The Group S₄