The Group D2d(face)

There are two different subgroups with Schoenflies-type D2d. They differ in the position of the two-fold rotational axes, which can pass trough the face centers or the edge centers. Up to M-symmetry there are 192 cubes which exactly have the symmetries of this subgroup. All cubes are generated in 19 moves or less. A few examples are listed below.

The table gives the number of cubes up to M-symmetry which exactly have D2d(face)-symmetry, the average maneuver length is 15.03:

Distance
Number
Distance
Number
0f
0
11f
9
1f
0
12f
28
2f
0
13f
22
3f
0
14f
18
4f
0
15f
5
5f
0
16f
14
6f
0
17f
28
7f
0
18f
44
8f
1
19f
12
9f
0
20f
0
10f
11

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file D2df.zip.
Name
 shortest maneuver with exactly this symmetry
Generator
 U R L F2 B2 R' L' U (8f*)
Name
Generator
 L2 R2 D2 B2 F2 U' B2 F2 L2 R2 U' (11f*)
Name
Generator
 B D' F2 L' R' F D' L R U' B L' R' B2 U' F D' U' (18f*)
Name
Generator
 D2 U' L2 F' L2 D2 U2 R2 B' R2 U' (11f*)
Name
Generator
 U R2 F L2 R2 F2 L2 R2 F' R2 U' (11f*)
Name
Generator
 D' L2 F2 U2 B2 R2 U2 L2 D U' R2 U' (12f*)
Name
Generator
 B2 L' R' B L2 D U' F2 L' D U F D U' B F' U' (17f*)
Name
Generator
 B2 L2 R2 F2 D U2 F2 D2 U2 F2 L2 R2 U' (13f*)
Name
Generator
 D' U' R B2 F2 R' U2 B F' R2 U R2 B F' U2 R' (16f*)
Name
Generator
 L2 U2 B2 F2 U2 R2 U' L' R D2 U2 L R' U' (14f*)
Name
Generator
 L2 D' F2 L' F R D R B U' F' L' R U2 L B R' U2 (18f*)
Name
Generator
 L2 R2 U2 B2 F2 U R2 B2 F2 U2 B2 F2 U2 R2 U' (15f*)
Name
Generator
 U' L D2 L2 B F' U B2 F2 D' B' F R2 U2 R' U' (16f*)
Name
Generator
 U' L R D2 B2 F2 U2 L' R' U' (10f*)
Name
Generator
 U' L D L2 F' U R B2 F2 L' D' F R2 U' R' U' (16f*)

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© 2017  Herbert Kociemba