The Group D_3d

There are 12 cubes, which exactly have the symmetries of this subgroup which is characterized as the largest subgroup of M which maps a long diagonal (here the URF-DLB diagonal) onto itself. All cubes are generated in 19 moves or less and are selfinverse.

The next table gives the number of cubes up to M-symmetry which exactly have D_3d-symmetry, the average maneuver length is 17.8:

Distance	Number	Distance	Number
14f	1	18f	4
17f	3	19f	4

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file D3d.zip.

Name	shortest maneuver with exactly this symmetry
Generator	U L D U L' D' U' R B2 U2 B2 L' R' U' (14f*)

Name
Generator	U2 L2 R2 F' R' U B R' D B2 F2 L' B' D R B' U' (17f*)

Name
Generator	L2 B' U B' D' B F D2 F' U' B L2 U' F2 U2 R2 U' (17f*)

Name
Generator	D2 B' U2 L B2 D2 U F2 D B2 R' B2 U B D' F2 U' (17f*)

Name
Generator	D F R' U2 B2 U2 R' U' L R' B U' R2 F2 R2 U' R' U' (18f*)

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

Name
Generator	U L U B' U' L R2 U2 B U' L' U' B F2 R2 B' F2 U' (18f*)

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

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

Name
Generator	L R2 D' B F2 L2 R' B' D2 R2 D' U' R' D2 B D U' F' U' (19f*)

Name
Generator	D' U2 B2 L2 R2 B' R D U' L R' D B2 F2 R B' F' R U' (19f*)

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

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The Group D3d

The Group D_3d