The Group C_4h

Up to M-symmetry there are 704 cubes which exactly have the symmetries of this subgroup. There are 32 cubes up to M-symmetry which need 20 moves. 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 C_4h-symmetry, the average maneuver length is 16.97:

Distance	Number	Distance	Number
0f	0	11f	10
1f	0	12f	28
2f	1	13f	35
3f	0	14f	20
4f	0	15f	34
5f	0	16f	87
6f	3	17f	95
7f	1	18f	184
8f	3	19f	171
9f	0	20f	32
10f	0

If you are interested in a list of all optimal maneuvers, they are included in the file C4h.zip. The 28 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

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

Name
Generator	L2 R2 U2 B2 F2 U B2 F2 U2 L2 R2 U' (12f*)

Name
Generator	F2 L R D2 F L2 R2 D2 U2 F' R2 U2 L' R' (14f*)

Name
Generator	D' L2 U2 R2 B2 D U B2 L2 U2 R2 U' (12f*)

Name
Generator	L2 R2 U2 B2 F2 D B2 F2 U2 L2 R2 U' (12f*)

Name
Generator	U' L2 U2 L2 B2 D U B2 R2 U2 R2 U' (12f*)

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

Name
Generator	L2 R2 F2 L2 D U B2 R2 D' U' F2 R2 U2 (13f*)

Name
Generator	B2 L2 D' U' B2 R2 D U F2 R2 U2 (11f*)

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The Group C4h

The Group C_4h