Rubik's Cube Group: Number of Elements With Each Order

Question

In the Rubik's Cube group, for each order, how many elements are there that have that order?

Answer 1

According to https://www.jaapsch.net/puzzles/cubic3.htm#p34, the counts are below. The most common order is 60, with about 10.6% of elements having that order. The least common order (besides the identity) is 11, with only about 0.000000103% of elements having that order. The largest order of any element in the Rubik's Cube group is 1260.

Order	Number of Elements	Fraction of Group
1	1	0.0000000000000000023%
2	170,911,549,183	0.000000395%
3	33,894,540,622,394	0.000078365%
4	4,346,957,030,144,250	0.010050302%
5	133,528,172,514,624	0.000308721%
6	140,621,059,298,755,000	0.325120338%
7	153,245,517,148,800	0.000354308%
8	294,998,638,981,939,000	0.682046187%
9	55,333,752,398,428,800	0.127933386%
10	65,250,897,836,352,100	0.150862140%
11	44,590,694,400	0.000000103%
12	2,330,232,827,455,550,000	5.387572022%
14	23,298,374,383,021,400	0.053866579%
15	14,385,471,333,209,800	0.033259665%
16	150,731,886,270,873,000	0.348496890%
18	520,622,849,158,124,000	1.203696499%
20	198,732,245,664,927,000	0.459475240%
21	39,337,151,559,333,100	0.090948739%
22	927,085,127,270,400	0.002143450%
24	3,293,932,519,796,240,000	7.615676201%
28	97,419,760,907,673,600	0.225237569%
30	2,033,284,208,966,740,000	4.701017421%
33	15,874,019,662,233,600	0.036701236%
35	65,526,218,912,563,200	0.151498691%
36	3,186,202,597,973,170,000	7.366601213%
40	1,136,258,380,254,410,000	2.627065325%
42	737,199,776,831,097,000	1.704429208%
44	100,120,377,950,208,000	0.231481481%
45	197,329,441,659,727,000	0.456231912%
48	911,497,647,410,380,000	2.107411399%
55	4,854,321,355,161,600	0.011223345%
56	671,205,306,846,412,000	1.551847905%
60	4,601,524,692,892,920,000	10.638870675%
63	264,371,433,705,308,000	0.611235119%
66	404,051,175,250,329,000	0.934179101%
70	353,490,834,273,730,000	0.817281993%
72	1,681,092,660,339,670,000	3.886739418%
77	187,238,109,413,376,000	0.432900433%
80	13,349,383,726,694,400	0.030864198%
84	1,697,725,818,678,060,000	3.925195806%
90	1,925,069,051,617,380,000	4.450820554%
99	104,367,909,135,974,000	0.241301908%
105	232,824,419,423,354,000	0.538297424%
110	4,854,321,355,161,600	0.011223345%
112	128,726,200,221,696,000	0.297619048%
120	1,947,044,011,463,140,000	4.501627356%
126	425,696,352,223,887,000	0.984223435%
132	637,129,677,864,960,000	1.473063973%
140	223,125,413,717,606,000	0.515873016%
144	714,192,029,378,150,000	1.651234568%
154	187,238,109,413,376,000	0.432900433%
165	213,590,139,627,110,000	0.493827160%
168	1,050,269,239,266,500,000	2.428255710%
180	2,453,889,005,738,310,000	5.673469019%
198	759,701,292,082,790,000	1.756453423%
210	1,339,232,732,046,950,000	3.096348448%
231	374,476,218,826,752,000	0.865800866%
240	407,156,203,664,179,000	0.941358025%
252	689,877,080,447,385,000	1.595017637%
280	68,653,973,451,571,200	0.158730159%
315	99,309,879,652,515,800	0.229607584%
330	213,590,139,627,110,000	0.493827160%
336	257,452,400,443,392,000	0.595238095%
360	571,019,888,909,352,000	1.320216049%
420	961,155,628,321,996,000	2.222222222%
462	374,476,218,826,752,000	0.865800866%
495	174,755,568,785,817,000	0.404040404%
504	238,381,852,262,400,000	0.551146384%
630	395,380,140,162,416,000	0.914131393%
720	120,144,453,540,249,000	0.277777778%
840	240,288,907,080,499,000	0.555555556%
990	174,755,568,785,817,000	0.404040404%
1260	51,490,480,088,678,400	0.119047619%