The Group D₂ (face)

Silviu Radu did the analysis of the 294912 different cubes of this subgroup for the cubes with distance less than 20. Here the the table with the distances, the average distance is 15.69, what is quite low:

Distance	Number	Distance	Number
0f	1	11f	7,776
1f	0	12f	17,780
2f	9	13f	20,250
3f	0	14f	28,698
4f	63	15f	23,343
5f	24	16f	45,731
6f	424	17f	66,418
7f	246	18f	57,353
8f	2,000	19f	16,711
9f	1,593	20f	210
10f	6,282

Up to M-symmetry there are 23356 cubes which exactly have this symmetry. All cubes can be solved within 20 moves and there are only 4 cubes up to M-symmetry which need 20 moves. It is interesting to observe that all 4 cubes have an antisymmetry defined by the complement of D₂ (face) in D_2h(face). They are displayed below, together with a few other examples of this class.

The next table gives the number of cubes up to M-symmetry which exactly have D₂ (face)-symmetry, the average maneuver length is 15.67 now:

Distance	Number	Distance	Number
0f	0	11f	641
1f	0	12f	1,428
2f	0	13f	1,608
3f	0	14f	2,185
4f	4	15f	1,850
5f	2	16f	3,708
6f	27	17f	5,370
7f	18	18f	4,456
8f	152	19f	1,256
9f	132	20f	4
10f	515

If you are interested in a list of all optimal maneuvers, they are included in the file D2f.zip. The four 20f*-maneuvers also are included in the file 20moves.zip.

Name	shortest maneuver with exactly this symmetry
Generator	R2 L2 F B (4f*)

Name
Generator	B L2 R2 D2 U2 B2 F' L B2 F2 D2 U2 L2 R' (14f*)

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

Name
Generator	R D R' U B' R' D' U R F D' R U' R' U2 (15f*)

Name
Generator	B' F L2 R2 B F' D U (8f*)

Name
Generator	B2 U R2 B D U2 R2 B R' D R2 D2 F' D' L' R' B' F' R' U' (20f*)

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

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

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

Cube display with AnimCubeJS

The Group D2 (face)

The Group D₂ (face)