The Group D4h

Up to M-symmetry there are 124 cubes, which exactly have the symmetries of this subgroup which plays an important part in the symmetry reduction of the Two-Phase-Algorithm coordinates. D4h is the largest subgroup of M which preserves the UD-axis. Up to M-symmetry all cubes but one are generated in 19 moves or less. One cube needs 20 moves. It is the 4-spot combined with superflip. It is interesting to observe, that all 124 cubes are selfinverse.

The table gives the number of cubes up to M-symmetry which exactly have D4h-symmetry, the average maneuver length is 14.9:

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

If you are interested in a list of all optimal maneuvers, download the file D4h.zip. The 20f*-maneuver also is included in the file 20moves.zip.
Name
 shortest maneuver with exactly this symmetry
Generator
 D2 U2 (2f*)
Name
Generator
 D B2 F2 D' U L2 R2 U' (8f*)
Name
Generator
 B' F' D' U' L' R' B' F' D' U' L' R' (12f*)
Name
Generator
 D B2 L2 B2 D U' R2 F2 R2 U' (10f*)
Name
Generator
 D2 L2 F2 L2 R2 F2 R2 U2 (8f*)
Name
Generator
 B F D U L R B' F' D' U' L' R' (12f*)
Name
Generator
 L2 F2 R2 B2 F2 R2 F2 R2 (8f*)
Name
Generator
 D2 B2 F2 L2 R2 U2 (6f*)
Name
Generator
 L2 R2 D U' L2 R2 D U' (8f*)
Name
Generator
 D F2 R2 F2 D' U R2 F2 R2 U' (10f*)
Name
Generator
 L2 F2 L2 R2 F2 R2 (6f*)
Name
Generator
 D B2 F2 D U' L2 R2 U' (8f*)
Name
Generator
 L2 R2 D' U L2 R2 D U' (8f*)
Name
Generator
 B2 F2 L2 R2 (4f*)
Name
Generator
 B F D U L' R' B F D' U' L' R' (12f*)
Name
Generator
 U2 L2 R2 D U' L2 R2 D' U' (9f*)
Name
Generator
 U2 B2 F2 L2 R2 U2 (6f*)
Name
Generator
 B' F' D' U' L R B F D' U' L' R' (12f*)
Name
Generator
 U R2 B F L R B F D U R2 U' (12f*)
Name
Generator
 U R2 B' F' L' R' B' F' D' U' R2 U' (12f*)

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© 2017  Herbert Kociemba