The Group C₁

The Group C₁ is the whole cube group which contains 43,252,003,274,489,856,000 elements. Dan Hoey showed in 1994 that this group contains 901,083,404,981,813,616 elements up to M-symmetry. Mike Godfrey and I showed in 2005 that there are 450,541,810,590,509,978 elements up to M-symmetry and antisymmetry.

We have the following table (see sequence A080601 in the On-Line Encyclopedia of Integer Sequences) for the distances in this subgroup:

Distance	Number	Distance	Number
0f	1	11f	3,063,288,809,012
1f	18	12f	40,374,425,656,248
2f	243	13f	531,653,418,284,628
3f	3,240	14f	6,989,320,578,825,358
4f	43,239	15f	91,365,146,187,124,313
5f	574,908	16f	unknown
6f	7,618,438	17f	unknown
7f	100,803,036	18f	unknown
8f	1,332,343,288	19f	unknown
9f	17,596,479,795	20f	unknown
10f	232,248,063,316	21f	0

Because we have the tables for all other symmetry groups except C₁ which exactly have a given symmetry, we can compute the table for the number of cubes wich are symmetric:

Distance	Number	Distance	Number
0f	1	11f	9,732,164
1f	18	12f	35,024,904
2f	51	13f	122,054,340
3f	312	14f	436,197,214
4f	1,335	15f	1,763,452,505
5f	4,380	16f	8,035,307,127
6f	17,782	17f	37,542,012,922
7f	70,188	18f	95,387,902,305
8f	229,336	19f	21,267,102,443
9f	851,139	20f	1,091,994
10f	2,989,204	21f	0

This are altogether 164,604,041,664 cubes. By subtracting the second table from the first we can compute the number of cubes which exactly have C₁-symmetry (and hence are unsymmetric):

Distance	Number	Distance	Number
0f	0	11f	3,063,279,076,848
1f	0	12f	40,374,390,631,344
2f	192	13f	531,653,296,230,288
3f	2,928	14f	6,989,320,142,628,144
4f	41,904	15f	91,365,144,423,671,808
5f	570,528	16f	unknown
6f	7,600,656	17f	unknown
7f	100,732,848	18f	unknown
8f	13,32,113,952	19f	unknown
9f	17,595,628,656	20f	unknown
10f	232,245,074,112	21f	0

If we want to know the number of cubes which exactly have C₁-symmetry mod M, we have to divide all numbers in the last table by 48.

It is interesting to notice that the the probability to get a symmetric cube by random is 164,604,041,664 / 43,252,003,274,489,856,000 which is only about 3.81*10^-9.

There are myriads of nice unsymmetric cubes. Some examples are given below.

Name	shortest maneuver with exactly this symmetry
Generator	U R (2f*)

Name
Generator	L2 R F2 R' U' L' U B F2 L2 D' L D' B' (14f*)

Name
Generator	L U B2 L2 D2 B U B' D' R' B' F' U2 B2 U R (16f*)

Name
Generator	R F U F2 U2 R' B' F2 D' L' D2 L2 B D2 (14f*)

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

Name
Generator	L2 D2 U2 L' U2 L' R2 D2 U2 R' U2 R' (12f*)

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The Group C1

The Group C₁