The Group D_2d(edge)

Up to M-symmetry there are 1472 cubes which exactly have the symmetries of this subgroup. There are 17 cubes up to M-symmetry which need 20 moves, 5 cubes 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_2d(edge)-symmetry, the average maneuver length is 16.58:

Distance	Number	Distance	Number
0f	0	11f	22
1f	0	12f	41
2f	0	13f	57
3f	0	14f	103
4f	0	15f	103
5f	0	16f	208
6f	0	17f	242
7f	3	18f	389
8f	8	19f	256
9f	13	20f	17
10f	10

If you are interested in a list of all optimal maneuvers, they are included in the file D2de.zip. The eleven 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	U F2 B2 D2 F2 B2 U (7f*)

Name
Generator	L' D R2 F' L2 U L R U' L2 B R2 D' R' (14f*)

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

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

Name
Generator	B2 F U2 L2 U L B2 U' F' R D L2 B' D' B2 D2 L2 R' (18f*)

Name
Generator	D' F R2 D' U F2 L' B F L2 D L' R D U' R' U' (17f*)

Name
Generator	B2 R' D2 B2 R' B D2 U2 B' U F2 L2 U F2 R' U2 R2 U' (18f*)

Name
Generator	U' L2 R2 U2 L2 R2 U' (7f*)

Name
Generator	D' B2 F2 D U L2 R2 U' (8f*)

Name
Generator	B' F' U2 B' F' L' R' U2 L' R' (10f*)

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

Name
Generator	B D' B F2 U' F' L' F D R' F2 R B D' B2 F2 D R' (18f*)

Name
Generator	R2 B F' L2 R2 B F' R2 U2 (9f*)

Name
Generator	U' L R' D' R2 B' F' L B2 D U' L2 F' D' U' R' U' (17f*)

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The Group D2d (edge)

The Group D_2d(edge)