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