The Group C_2h(a)

Silviu Radu did most of the analysis of the 589824 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.61:

Distance	Number	Distance	Number
0f	1	11f	1,768
1f	0	12f	2,916
2f	5	13f	6,002
3f	0	14f	12,426
4f	23	15f	19,795
5f	8	16f	46,029
6f	102	17f	108,816
7f	34	18f	239,765
8f	322	19f	149539
9f	137	20f	1636
10f	500

Up to M-symmetry there are 143552 cubes which exactly have this symmetry. All cubes can be solved within 20 moves and there are 350 cubes up to M-symmetry, which need 20 moves. All except 57 of these cubes are antisymmetric (that is the inverse is M-symmetric to the cube itself).

The next table gives the number of cubes up to M-symmetry which exactly have C_2h (a)-symmetry, the average maneuver length is 17.65 now:

Distance	Number	Distance	Number
0f	0	11f	423
1f	0	12f	622
2f	0	13f	1285
3f	0	14f	2489
4f	2	15f	4636
5f	2	16f	11153
6f	5	17f	26636
7f	7	18f	58874
8f	42	19f	36878
9f	33	20f	350
10f	115

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file C2ha.zip. The 293 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

A few other nice examples are listed below.

Name	shortest maneuver with exactly this symmetry
Generator	U' D F2 B2 (4f*)

Name
Generator	R2 B2 F2 R2 U' B2 F2 D2 L2 R2 U' (11f*)

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

Name
Generator	D2 L2 R2 D' B2 U2 L2 R2 U2 F2 U' (11f*)

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

Name
Generator	L' B F' D' U2 B2 D' B F' R' D U' F2 R2 (14f*)

Name
Generator	L' D' L' B' U L D U2 L D B U' B' U F2 U' (16f*)

Name
Generator	U2 R D' B L' U L2 R2 D' L F' U R' U2 (14f*)

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The Group C2h (a)

The Group C_2h(a)