The Group C_2v (a2)

Silviu Radu did most of the analysis of the 1,179,648 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.68:

Distance	Number	Distance	Number
0f	1	11f	1184
1f	2	12f	4284
2f	3	13f	9252
3f	8	14f	20858
4f	15	15f	32193
5f	36	16f	89617
6f	92	17f	223444
7f	140	18f	505801
8f	230	19f	289047
9f	167	20f	2780
10f	494

Up to M-symmetry there are 290880 cubes which exactly have this symmetry. All but 646 cubes up to M-symmetry can be solved with less than 20 moves. From these 646 cubes, 244 have antisymmetry.

The next table gives the number of cubes up to M-symmetry which exactly have C_2v (a2)-symmetry, the average maneuver length is 17.70 now:

Distance	Number	Distance	Number
0f	0	11f	280
1f	0	12f	960
2f	0	13f	2080
3f	2	14f	4549
4f	0	15f	7712
5f	8	16f	22012
6f	4	17f	55305
7f	33	18f	125363
8f	17	19f	71764
9f	34	20f	646
10f	111

If you are interested in a list of all optimal maneuvers, they are included in the file C2va2.zip.The 20 move maneuvers for the 445 cubes up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

Below are some examples of cubes with this kind of symmetry.

Name	shortest maneuver with exactly this symmetry
Generator	R2 L2 U2 (3f*)

Name
Generator	L F' U L' B' F U R' F U' R B F' U' (14f*)

Name
Generator	L F2 R2 D2 R D2 L' F2 R2 U2 R' U2 (12f*)

Name
Generator	L2 B L' R D2 U' B2 R2 B2 U' L' R F' R2 (14f*)

Name
Generator	D' B2 L' R' F U2 L R' F2 D' B F L D U' L R' (17f*)

Name
Generator	R U B2 D2 F2 D' B' U L' D' R2 D2 L2 U F U' (16f*)

Name
Generator	D2 F U' R2 D R' U2 R D' R2 U F' U2 (13f*)

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The Group C2v (a2)

The Group C_2v (a2)