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 D2h(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*)

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© 2017  Herbert Kociemba