The Group C_{2} (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:
Up to Msymmetry 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 Msymmetry and Mantisymmetry 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 Msymmetry which exactly have C_{2} (a)symmetry. To compute this number we used the following identity, where C_{2} (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_{2 }(a), which are all known except for C_{2 }(a) itself. C_{2} (a)Total = 8 C_{2} (a) + 4 C_{2v }
(a2) + 4 C_{2h} (a) + 4 S_{4} + 12 D_{2}
(face) + 4 C_{4} + 4 D_{2} (edge) + 4 C_{2v
} (a1) + 6 D_{2h}(face) + 2 D_{2d }(edge)
+ 2D_{2h }(edge) + 2C_{4h} + 2C_{4v} + 6 D_{2d
}(face) + The average maneuver length also is 17.49:


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© 2017 Herbert Kociemba 