The Group C_2v (b)

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

Up to M-symmetry there are 48128 cubes which exactly have this symmetry. All cubes can be solved within 20 moves. There are 94 cubes up to M-symmetry, which need 20 moves. All except 2 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_2v (b)-symmetry, the average maneuver length is 17.59:

Distance	Number	Distance	Number
0f	0	11f	58
1f	0	12f	183
2f	0	13f	293
3f	0	14f	671
4f	0	15f	1,491
5f	0	16f	4,381
6f	3	17f	10,329
7f	1	18f	20,873
8f	3	19f	9,696
9f	18	20f	94
10f	34

If you are interested in a list of all optimal maneuvers, they are included in the file C2vb.zip.The 20 move maneuvers for the 93 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	B2 R2 B2 R2 B2 R2 (6f*)

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

Name
Generator	U R B F L2 B' F' R' U' (9f*)

Name
Generator	L B U B2 F2 D' B' R' U B2 R2 D U F2 U' (15f*)

Name
Generator	D' L2 D U L2 U' (6f*)

Name
Generator	R' B' D2 F D F' U F' D U2 B U2 R D' F2 D R2 U' (18f*)

Name
Generator	L R2 D2 L R U2 F' U' F U' F R2 D B U' L2 B R' (18f*)

Name
Generator	B2 D' U' B2 L2 U' R F' U F2 U' F L2 R' U2 (15f*)

Name
Generator	L2 B' L2 R2 D2 B' L' D F2 U' R' F' L D F' U F' U' (18f*)

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

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

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

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The Group C2v (b)

The Group C_2v (b)