The Group D₄

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 D₄-symmetry, the average maneuver length is 14.52:

Distance	Number	Distance	Number
0f	0	11f	5
1f	0	12f	30
2f	1	13f	24
3f	0	14f	8
4f	0	15f	12
5f	0	16f	20
6f	4	17f	38
7f	5	18f	22
8f	5	19f	14
9f	1	20f	0
10f	3

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file D4.zip.

Name	shortest maneuver with exactly this symmetry
Generator	U D (2f*)

Name
Generator	B2 F2 L2 R2 U B2 F2 L2 R2 U' (10f*)

Name
Generator	L2 B2 F2 L R' F' L2 R2 D2 U2 F L' R' (13f*)

Name
Generator	B R' D F2 L' B D' U R' B L2 D' F R' U2 (15f*)

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

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

Name
Generator	F2 R2 D' U' R2 B2 L2 F2 D U F2 R2 (12f*)

Name
Generator	B2 L2 D' U' L2 F2 L2 F2 D U F2 R2 (12f*)

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The Group D4

The Group D₄