The Group C_2v (a1)

Silviu Radu did most of the analysis of the 65536 different cubes of this subgroup. Here the the table with the distances, the average distance is 16.98:

Distance	Number	Distance	Number
0f	1	11f	462
1f	2	12f	756
2f	1	13f	832
3f	0	14f	3,112
4f	1	15f	5,249
5f	4	16f	9,769
6f	4	17f	15,690
7f	10	18f	22,009
8f	26	19f	7,285
9f	9	20f	168
10f	146

Up to M-symmetry there are 15552 cubes which exactly have this symmetry. All but 39 cubes up to M-symmetry can be solved with less than 20 moves. 26 of the 20-move antipodes are selfinverse, for the other 13 cubes a point reflection at the cube center gives the inverse cubes. The 39 antipodes and six other examples are listed below.

The next table gives the number of cubes up to M-symmetry which exactly have C_2v (a1)-symmetry, the average maneuver length is 17.01 now:

Distance	Number	Distance	Number
0f	0	11f	92
1f	0	12f	134
2f	0	13f	130
3f	0	14f	704
4f	0	15f	1,249
5f	0	16f	2,347
6f	0	17f	3,735
7f	1	18f	5,340
8f	1	19f	1,757
9f	1	20f	39
10f	22

If you are interested in a list of all optimal maneuvers, they are included in the file C2va1.zip.The 20 move maneuvers for the 39 cubes up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip. Some examples are given below.

Name	shortest maneuver with exactly this symmetry
Generator	U R2 L2 U2 F2 B2 U' (7f*)

Name
Generator	B2 F2 D L2 R2 U' L R' D2 U2 L R' (12f*)

Name
Generator	B2 L2 B2 D R2 F2 R2 B2 U B2 F2 U F2 U' (14f*)

Name
Generator	D' B2 F2 U' B2 F2 U' F2 D2 U2 F2 U' (12f*)

Name
Generator	L2 D R2 F2 U B2 R U B' U2 B U' R F2 U R2 U' (17f*)

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

Name
Generator	B U' B2 U B' F R D L2 R2 D' F2 L2 F R' (15f*)

Name
Generator	D R2 D F2 L2 D F2 D' B F U2 L D U F' R2 U2 F' L' R' (20f*)

Name
Generator	D' U2 B2 R2 U R2 D' L R' D B2 U' L U2 B' R U' L2 F R' (20f*)

Name
Generator	U L2 D2 R2 F2 D' B2 U' B D F R' B L D2 U' F D' F2 R (20f*)

Name
Generator	L' F2 D' L2 R' D' L' F2 D2 U' L' F U' L2 D' B2 U2 L' F' U' (20f*)

Name
Generator	U R2 U L2 F2 L2 U' L2 U' L2 B2 U2 B F L' D' U' F2 L R (20f*)

Name
Generator	B2 D F2 U' B2 U R2 D' L R F D U R' D2 F2 L B F R2 (20f*)

Name
Generator	L B2 R2 D F' D2 R2 U B F L D L' F' U2 R' F U2 R2 U' (20f*)

Name
Generator	D' R2 D2 L2 F2 U B2 U' B D F R' B L D2 U' F D' F2 R (20f*)

Name
Generator	B2 U F2 D' B2 D L' R' B D U L' D2 B2 L' D2 L2 B F' U' (20f*)

Name
Generator	F2 L2 U2 F2 U' R2 D' F L' D' B' U' L2 D R' B' L R D2 R (20f*)

Name
Generator	B2 R2 F L2 U' R F2 L' U L' D' U R U2 R U R2 U2 F' U' (20f*)

Name
Generator	B' U2 R2 U R U2 R D' U L' U L' B2 R U' L2 B R2 F2 U' (20f*)

Name
Generator	U R2 D F2 U2 B2 L' R' B' D U' R2 B2 L B' F' U2 R2 U' R2 (20f*)

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The Group C2v (a1)

The Group C_2v (a1)