The Group C2h (b)

Silviu Radu did most of the analysis of the 98304 different cubes of this subgroup.

Up to M-symmetry there are 48116 cubes which exactly have this symmetry. All cubes can be solved within 20 moves and there are 71 cubes up to M-symmetry, which need 20 moves. All except 4 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 (b)-symmetry, the average maneuver length is 17.56:

Distance
Number
Distance
Number
0f
0
11f
72
1f
0
12f
192
2f
0
13f
329
3f
0
14f
671
4f
0
15f
1720
5f
0
16f
4360
6f
1
17f
10513
7f
1
18f
20776
8f
7
19f
9360
9f
10
20f
71
10f
33

If you are interested in a list of all optimal maneuvers, they are included in the file C2hb.zip.The 20 move maneuvers for the 69 cubes 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 R2 U D R2 D (6f*)
Name
Generator
R B' R2 D R D' R F2 L' U L B F2 R' (14f*)
Name
Generator
F2 R2 D R2 D U F2 D' R' D' F L2 F' D R U' (16f*)
Name
Generator
B L U' R D2 U' L D B' L B' L2 U' L' B' R U' (17f*)
Name
Generator
F D R' U' B' L' R' U L2 F L' B L B2 F D R' (17f*)
Name
Generator
R2 B2 U' B2 F2 U R' D U' B' L2 B D' U R' (15f*)

Cube display with AnimCubeJS

© 2017  Herbert Kociemba