The Group C₄

Silviu Radu did most of the analysis of the 147456 different cubes of this subgroup. Here the the table with the distances, the average distance is 16.95:

Distance	Number	Distance	Number
0f	1	11f	758
1f	6	12f	1,720
2f	9	13f	2,604
3f	0	14f	4,040
4f	1	15f	9,401
5f	12	16f	21,803
6f	50	17f	46,296
7f	42	18f	48,801
8f	56	19f	11,413
9f	33	20f	222
10f	188

Up to M-symmetry there are 36160 cubes which exactly have this symmetry. All but 39 cubes up to M-symmetry can be solved with less than 20 moves. 35 out of these 39 cubes have antisymmetry. All antipodes of this class are listed below, together with a few nice examples of this class.

The next table gives the number of cubes up to M-symmetry which exactly have C₄-symmetry, the average maneuver length is 16.97 now:

Distance	Number	Distance	Number
0f	0	11f	177
1f	1	12f	381
2f	1	13f	583
3f	0	14f	958
4f	0	15f	2,302
5f	2	16f	5,362
6f	8	17f	11,461
7f	7	18f	12,066
8f	7	19f	2,756
9f	7	20f	39
10f	42

If you are interested in a list of all optimal maneuvers, they are included in the file C4.zip.The 20 move maneuvers for the 37 cubes up to M-symmetry and M-antisymmetry are included in the file 20moves.zip.

Name	shortest maneuver with exactly this symmetry
Generator	D (1f*)

Name
Generator	F L' U2 B' R' D B U2 L F U' R' U (13f*)

Name
Generator	B2 F2 L2 R2 D B2 F2 L2 R2 (9f*)

Name
Generator	R2 U2 B' F D2 U2 L' F2 D' U L2 B' D2 B2 L R' U' (17f*)

Name
Generator	F R' B' D2 L' U' F L D2 B D R' (12f*)

Name
Generator	D B2 D' L2 U L' U F L F U' R' B2 D B U B F' D R' (20f*)

Name
Generator	B2 F2 U L2 B2 L R B L2 U' B L2 D' U' B' F' D2 U' B R2 (20f*)

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The Group C4

The Group C₄