The Group C_2h (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 C_2h (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*)

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The Group C2h (b)

The Group C_2h (b)