The Group C2h (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 C2h (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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© 2017  Herbert Kociemba