The Group C2 (a)

Silviu Radu managed to do a complete analysis of the 15,288,238,080 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.49:

Distance
Number
Distance
Number
0f
1
11f
2,602,404
1f
6
12f
9,295,556
2f
15
13f
30,949,920
3f
72
14f
101,692,042
4f
343
15f
363,402,817
5f
1,060
16f
1,333,918,995
6f
4,570
17f
4,550,677,966
7f
17,960
18f
7,774,262,733
8f
60,136
19f
1,120,268,931
9f
226,199
20f
51,094
10f
805,260

Up to M-symmetry there are 1,910,682,624 cubes which exactly have this symmetry and 5763 of these cubes need 20 moves. All other cubes can be solved in 19 moves or less. 3165 out of these 5763 cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 4464 20f*-cubes, which are included in the file 20moves.zip. We display only some nicer examples of this symmetry class here.

The next table gives the number of cubes up to M-symmetry which exactly have C2 (a)-symmetry. To compute this number we used the following identity, where C2 (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 C(a) itself.

C2 (a)Total = 8 C2 (a) + 4 C2v (a2) + 4 C2h (a) + 4 S4 + 12 D2 (face) + 4 C4 + 4 D2 (edge) + 4 C2v (a1) + 6 D2h(face) + 2 D2d (edge) + 2D2h (edge) + 2C4h + 2C4v + 6 D2d (face) +
6D4 + 3D4h + 4 T + 2 Th + Oh

The average maneuver length also is 17.49:

Distance
Number
Distance
Number
0f
0
11f
323,655
1f
0
12f
1,158,307
2f
0
13f
3,863,416
3f
8
14f
12,702,003
4f
34
15f
45,411,240
5f
123
16f
166,705,822
6f
507
17f
568,759,399
7f
2,182
18f
971,648,345
8f
7,210
19f
139,966,952
9f
28,017
20f
5,763
10f
99,641
Name
shortest maneuver with exactly this symmetry
Generator
L R U2 (3f*)
Name
Generator
B2 U' L R' B D' U L2 B2 R' B' F U' R2 (14f*)
Name
Generator
D U' B F D' B2 F2 U B F D U' (12f*)
Name
Generator
U2 B F' L2 R2 B F' U (8f*)
Name
Generator
F2 L' R D2 F' D2 U2 R2 D U' F2 L U2 B F' R2 (16f*)
Name
Generator
U' L' R D2 U2 L R' D' (8f*)
Name
Generator
U' B2 F2 D' (4f*)
Name
Generator
D U2 B2 F2 U' (5f*)

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© 2017  Herbert Kociemba