# The Group Cs (a)

Silviu Radu managed to do a complete analysis of the 18,345,885,696 different cubes of this subgroup which has the highest number of 20f*-maneuvers from all symmetric subgroups. Here the the table with the distances, the average distance is 17.87:

 Distance Number Distance Number 0f 1 11f 324,496 1f 4 12f 1,019,512 2f 13 13f 4,128,898 3f 48 14f 20,060,486 4f 159 15f 103,107,447 5f 464 16f 607,357,293 6f 1,554 17f 3,601,456,840 7f 4,758 18f 11,082,306,785 8f 13,810 19f 2,925,751,099 9f 40,353 20f 197,592 10f 114,084

Up to M-symmetry there are 2,292,846,080 cubes which exactly have this symmetry and of these there are 23793 cubes which need 20 moves. All other cubes can be solved in 19 moves or less. 7391 out of these 23793 cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 15592 20f*-cubes, which are included in the file 20moves.zip. We display only some examples of this huge symmetry class here.

The next table gives the number of cubes up to M-symmetry which exactly have C(a)-symmetry. To compute this number we used the following identity, where C(a)Total is the table above and the right side hold the tables for the number of cubes mod M with exactly the symmetry of the subgroups of C(a), which are all known except for Cs (a) itself.

C(a)Total = 8 C(a) + 4 C2h (a) + 4 C2v (b) + 8 C2v (a2) + 6 D2h (face) + 2 D2h (edge) +
4 D2d (edge) + 2 C4h + 4 C4v + 3 D4h + 2 Th + Oh

The average maneuver length also is 17.87:

 Distance Number Distance Number 0f 0 11f 40,016 1f 0 12f 125,984 2f 1 13f 513,072 3f 4 14f 2,501,038 4f 17 15f 12,877,423 5f 48 16f 75,889,571 6f 176 17f 450,107,871 7f 555 18f 1,385,122,357 8f 1,662 19f 365,623,453 9f 4,977 20f 23,793 10f 14,062
 Name shortest maneuver with exactly this symmetry Generator F2 R2 (2f*)
 Name Generator L R2 D' L B2 F2 R' B R2 F2 L D' L R2 B2 R' (16f*)
 Name Generator D2 B2 U2 B2 D' F' L2 F R' B2 R B' U2 B U' (15f*)
 Name Generator R2 B2 R' D B F2 D' B2 R2 B' R' D U' L' U F2 U' (17f*)
 Name Generator R' B' R' B D B2 F2 U' L2 F2 L D' F' D2 F' L' (16f*)
 Name Generator U2 B' R2 D2 F2 U' L' U2 F2 D2 U2 R2 F' R' F D' U' (17f*)
 Name Generator R B2 F U' F L' B' R2 D U' L B' L' U' L' U (16f*)

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© 2017 Herbert Kociemba