The Group D_2h(face)

Up to M-symmetry there are 1982 cubes which exactly have the symmetries of this subgroup. There are 25 cubes up to M-symmetry which need 20 moves, 9 of them also have antisymmetry. All other cubes can be solved within 19 moves or less. Some examples are listed below.

The table gives the number of cubes up to M-symmetry which exactly have D_2h(face)-symmetry, the average maneuver length is 16.24:

Distance	Number	Distance	Number
0f	0	11f	0
1f	0	12f	45
2f	0	13f	106
3f	0	14f	378
4f	2	15f	172
5f	0	16f	161
6f	11	17f	252
7f	0	18f	568
8f	20	19f	240
9f	0	20f	25
10f	2

If you are interested in a list of all optimal maneuvers, they are included in the file D2hf.zip. The seventeen 20f*-maneuvers up to M-symmetry and M-antisymmetry also are included in the file 20moves.zip.

Name	shortest maneuver with exactly this symmetry
Generator	B2 D2 U2 F2 (4f*)

Name
Generator	L2 R2 D' B2 U2 L' R F L2 R2 B D2 F2 L R' U' (16f*)

Name
Generator	R D2 B2 U2 B' R2 U' L' F' D' U R2 B D2 R B2 R2 F2 U' (19f*)

Name
Generator	R2 U' F2 U2 F2 L R F2 U2 F2 U2 L' R' U' R2 (15f*)

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

Name
Generator	U2 L2 R2 D' R2 F2 U2 B2 R2 U2 R2 D U' R2 U' (15f*)

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

Name
Generator	R B2 L2 F2 U2 L2 F2 R2 U2 L' D' U' R2 D' U' (15f*)

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

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

Name
Generator	D2 R2 F2 L2 R2 F2 D2 U2 R2 U2 (10f*)

Name
Generator	F2 U2 B2 F2 U2 F2 (6f*)

Name
Generator	B2 L B2 F2 D2 U2 L2 R' F2 D U2 B2 F2 L2 R2 U' (16f*)

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The Group D2h (face)

The Group D_2h(face)