The Group C_{i} 

With the help of GAP and methods similar to the methods used in Cube Explorer to find optimal maneuvers, Silviu Radu managed in a very ingenious way to do a complete analysis of the 45,864,714,240 different cubes . Here the the table with the distances, the average distance is 17.86:
Up to Msymmetry there are 1,910,931,706 cubes which exactly have the symmetry of this subgroup. All of them can be solved within 20 moves, 10540 up to Msymmetry need exactly 20 moves. Up to Msymmetry and Mantisymmetry it are 7188 20f*cubes, which are included in the file 20moves.zip. Below are some nice examples for cubes, which exactly have C_{i} symmetry. The next table gives the number of cubes up to Msymmetry which exactly have C_{i}symmetry. To compute this number we used the following identity, where C_{i}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_{i}, which are all known except for C_{i} itself. C_{i}Total = 24 C_{i}+ 12 C_{2h }(b)
+ 12 C_{2h }(a) + 6 D_{2h}(face) + 6 D_{2h }(edge)
+ 6 C_{4h} + The average maneuver length also is 17.86:


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© 2017 Herbert Kociemba 