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Small subgroups and cosets

Submitted by brac37 on Sat, 10/16/2010 - 18:29.
Hello all, I am making a program to scan subgroups and coset groups of Rubik's cube. For testing purposes, I scanned the square subgroup. It appeared that the corner coordinate of the square subgoup always satisfies a multiple of four antisymmetries (half of which are symmetries). Below follow the results. I computed modulo counts for all 420 antisymmetry subgroups, of which I choose six to display. 
Square group table for FTM:
Level   |        | SO     | GO     | inv    | SO+inv | GO+inv |
--------+--------+--------+--------+--------+--------+--------+
    0   |      1 |      1 |      1 |      1 |      1 |      1 | 
    1   |      6 |      1 |      1 |      6 |      1 |      1 | 
    2   |     27 |      2 |      2 |     15 |      2 |      2 | 
    3   |    120 |      6 |      5 |     72 |      4 |      4 | 
    4   |    519 |     24 |     18 |    267 |     16 |     13 | 
    5   |   1932 |     86 |     56 |   1026 |     51 |     38 | 
    6   |   6484 |    280 |    162 |   3292 |    156 |     97 | 
    7   |  20310 |    859 |    482 |  10332 |    458 |    277 | 
    8   |  55034 |   2322 |   1258 |  27650 |   1227 |    695 | 
    9   | 113892 |   4878 |   2627 |  57642 |   2566 |   1467 | 
   10   | 178495 |   7618 |   4094 |  90383 |   4087 |   2265 | 
   11   | 179196 |   7702 |   4137 |  90954 |   4114 |   2389 | 
   12   |  89728 |   3979 |   2231 |  47054 |   2428 |   1402 | 
   13   |  16176 |    852 |    548 |   8724 |    566 |    424 | 
   14   |   1488 |    186 |    114 |   1194 |    175 |    103 | 
   15   |    144 |     16 |     16 |    108 |     16 |     16 | 

Square group table for STM:
Level   |        | SO     | GO     | inv    | SO+inv | GO+inv |
--------+--------+--------+--------+--------+--------+--------+
    0   |      1 |      1 |      1 |      1 |      1 |      1 | 
    1   |      9 |      2 |      2 |      9 |      2 |      2 | 
    2   |     51 |      4 |      4 |     27 |      3 |      3 | 
    3   |    265 |     16 |     15 |    157 |     12 |     12 | 
    4   |   1230 |     60 |     48 |    648 |     39 |     33 | 
    5   |   4767 |    214 |    145 |   2463 |    122 |     91 | 
    6   |  14868 |    646 |    392 |   7581 |    360 |    235 | 
    7   |  33863 |   1432 |    798 |  17171 |    764 |    459 | 
    8   |  70194 |   2938 |   1566 |  35439 |   1552 |    870 | 
    9   | 125616 |   5389 |   2923 |  63468 |   2830 |   1617 | 
   10   | 178672 |   7686 |   4114 |  90740 |   4162 |   2326 | 
   11   | 152928 |   6524 |   3484 |  77496 |   3467 |   1985 | 
   12   |  68848 |   3078 |   1718 |  36278 |   1921 |   1097 | 
   13   |  11328 |    670 |    454 |   6366 |    481 |    375 | 
   14   |    912 |    152 |     88 |    876 |    152 |     88 |
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More will follow

Submitted by brac37 on Sat, 10/16/2010 - 18:46.
More subgroups will follows, but small subgroups first since the program does not use symmetry reduction yet and neither routines to use the harddisk as memory (by way of a quartary table for all (anti)symmetry-reduced positions). 
Square group table for FTM, cumulative:
Level   |        | SO     | GO     | inv    | SO+inv | GO+inv |
--------+--------+--------+--------+--------+--------+--------+
    0   |      1 |      1 |      1 |      1 |      1 |      1 | 
    1   |      7 |      2 |      2 |      7 |      2 |      2 | 
    2   |     34 |      4 |      4 |     22 |      4 |      4 | 
    3   |    154 |     10 |      9 |     94 |      8 |      8 | 
    4   |    673 |     34 |     27 |    361 |     24 |     21 | 
    5   |   2605 |    120 |     83 |   1387 |     75 |     59 | 
    6   |   9089 |    400 |    245 |   4679 |    231 |    156 | 
    7   |  29399 |   1259 |    727 |  15011 |    689 |    433 | 
    8   |  84433 |   3581 |   1985 |  42661 |   1916 |   1128 | 
    9   | 198325 |   8459 |   4612 | 100303 |   4482 |   2595 | 
   10   | 376820 |  16077 |   8706 | 190686 |   8569 |   4860 | 
   11   | 556016 |  23779 |  12843 | 281640 |  12683 |   7249 | 
   12   | 645744 |  27758 |  15074 | 328694 |  15111 |   8651 | 
   13   | 661920 |  28610 |  15622 | 337418 |  15677 |   9075 | 
   14   | 663408 |  28796 |  15736 | 338612 |  15852 |   9178 | 
   15   | 663552 |  28812 |  15752 | 338720 |  15868 |   9194 | 

Square group table for STM, cumulative:
Level   |        | SO     | GO     | inv    | SO+inv | GO+inv |
--------+--------+--------+--------+--------+--------+--------+
    0   |      1 |      1 |      1 |      1 |      1 |      1 | 
    1   |     10 |      3 |      3 |     10 |      3 |      3 | 
    2   |     61 |      7 |      7 |     37 |      6 |      6 | 
    3   |    326 |     23 |     22 |    194 |     18 |     18 | 
    4   |   1556 |     83 |     70 |    842 |     57 |     51 | 
    5   |   6323 |    297 |    215 |   3305 |    179 |    142 | 
    6   |  21191 |    943 |    607 |  10886 |    539 |    377 | 
    7   |  55054 |   2375 |   1405 |  28057 |   1303 |    836 | 
    8   | 125248 |   5313 |   2971 |  63496 |   2855 |   1706 | 
    9   | 250864 |  10702 |   5894 | 126964 |   5685 |   3323 | 
   10   | 429536 |  18388 |  10008 | 217704 |   9847 |   5649 | 
   11   | 582464 |  24912 |  13492 | 295200 |  13314 |   7634 | 
   12   | 651312 |  27990 |  15210 | 331478 |  15235 |   8731 | 
   13   | 662640 |  28660 |  15664 | 337844 |  15716 |   9106 | 
   14   | 663552 |  28812 |  15752 | 338720 |  15868 |   9194 |
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Phase 1 cube without equator

Submitted by brac37 on Mon, 10/25/2010 - 18:14.
I computed the Phase 1 cube without equator. If you do not know what coset cube that is, see below. The Phase 1 cube is used in Cube Explorer.

EDIT: the cubes are "partially void", since some centers are equal. This is however not counted in the number of positions: the frame stays on its place. But since every symmetry on the upper face coset cube (symmetries that keep the upper face on place) and on the phase 1 coset cube without equator (symmetries that keep the UD axis on place) does nothing with the clean cube, the "partially void" results can be found in the SO column. 
Cube Explorer Phase 1 without equator coset table for QTM:
Level |           | SO        | GO        | cumul     | SO cumul  | GO cumul  |
------+-----------+-----------+-----------+-----------+-----------+-----------+
  0   |         1 |         1 |         1 |         1 |         1 |         1 | 
  1   |         4 |         1 |         1 |         5 |         2 |         2 | 
  2   |        34 |         5 |         3 |        39 |         7 |         5 | 
  3   |       312 |        42 |        24 |       351 |        49 |        29 | 
  4   |      2724 |       346 |       174 |      3075 |       395 |       203 | 
  5   |     24212 |      3041 |      1538 |     27287 |      3436 |      1741 | 
  6   |    215069 |     26934 |     13493 |    242356 |     30370 |     15234 | 
  7   |   1845578 |    230819 |    115615 |   2087934 |    261189 |    130849 | 
  8   |  14345836 |   1793570 |    897527 |  16433770 |   2054759 |   1028376 | 
  9   |  78807214 |   9851712 |   4929409 |  95240984 |  11906471 |   5957785 | 
 10   | 150911400 |  18865030 |   9440773 | 246152384 |  30771501 |  15398558 | 
 11   |  30922916 |   3865833 |   1938842 | 277075300 |  34637334 |  17337400 | 
 12   |     61340 |      7725 |      4187 | 277136640 |  34645059 |  17341587 | 

Cube Explorer Phase 1 without equator coset table for FTM:
Level |           | SO        | GO        | cumul     | SO cumul  | GO cumul  |
------+-----------+-----------+-----------+-----------+-----------+-----------+
  0   |         1 |         1 |         1 |         1 |         1 |         1 | 
  1   |         4 |         1 |         1 |         5 |         2 |         2 | 
  2   |        50 |         7 |         5 |        55 |         9 |         7 | 
  3   |       592 |        78 |        44 |       647 |        87 |        51 | 
  4   |      7028 |       887 |       457 |      7675 |       974 |       508 | 
  5   |     84364 |     10568 |      5326 |     92039 |     11542 |      5834 | 
  6   |    973249 |    121745 |     61039 |   1065288 |    133287 |     66873 | 
  7   |  10176130 |   1272231 |    636817 |  11241418 |   1405518 |    703690 | 
  8   |  74885232 |   9361372 |   4684044 |  86126650 |  10766890 |   5387734 | 
  9   | 167658384 |  20958654 |  10488780 | 253785034 |  31725544 |  15876514 | 
 10   |  23348828 |   2919151 |   1464814 | 277133862 |  34644695 |  17341328 | 
 11   |      2778 |       364 |       259 | 277136640 |  34645059 |  17341587 | 

Cube Explorer Phase 1 without equator coset table for STM:
Level |           | SO        | GO        | cumul     | SO cumul  | GO cumul  |
------+-----------+-----------+-----------+-----------+-----------+-----------+
  0   |         1 |         1 |         1 |         1 |         1 |         1 | 
  1   |         6 |         2 |         2 |         7 |         3 |         3 | 
  2   |        84 |        14 |        12 |        91 |        17 |        15 | 
  3   |      1225 |       162 |       100 |      1316 |       179 |       115 | 
  4   |     18174 |      2309 |      1214 |     19490 |      2488 |      1329 | 
  5   |    265216 |     33291 |     16841 |    284706 |     35779 |     18170 | 
  6   |   3618936 |    452728 |    227033 |   3903642 |    488507 |    245203 | 
  7   |  39762838 |   4971148 |   2487880 |  43666480 |   5459655 |   2733083 | 
  8   | 173506206 |  21689474 |  10852557 | 217172686 |  27149129 |  13585640 | 
  9   |  59917956 |   7490164 |   3752834 | 277090642 |  34639293 |  17338474 | 
 10   |     45998 |      5766 |      3113 | 277136640 |  34645059 |  17341587 | 

Cube Explorer Phase 1 without equator coset table for ATM:
Level |           | SO        | GO        | cumul     | SO cumul  | GO cumul  |
------+-----------+-----------+-----------+-----------+-----------+-----------+
  0   |         1 |         1 |         1 |         1 |         1 |         1 | 
  1   |         6 |         2 |         2 |         7 |         3 |         3 | 
  2   |       132 |        22 |        16 |       139 |        25 |        19 | 
  3   |      2953 |       394 |       218 |      3092 |       419 |       237 | 
  4   |     67594 |      8557 |      4364 |     70686 |      8976 |      4601 | 
  5   |   1538804 |    192763 |     96665 |   1609490 |    201739 |    101266 | 
  6   |  30140760 |   3768615 |   1885684 |  31750250 |   3970354 |   1986950 | 
  7   | 204214090 |  25527928 |  12773200 | 235964340 |  29498282 |  14760150 | 
  8   |  41171932 |   5146731 |   2581406 | 277136272 |  34645013 |  17341556 | 
  9   |       368 |        46 |        31 | 277136640 |  34645059 |  17341587 |
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Upper face coset table (not upper layer)

Submitted by brac37 on Sun, 10/24/2010 - 17:30.
The following table are the cosets of the group of monochrome upper faces. One can also see the cosets as 3x3x3 cubes with one red face and five grey faces. 
Monochrome upper face coset table for QTM:
Level     |          | SO       | GO       | cumul    | SO cumul | GO cumul |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |        8 |        2 |        1 |        9 |        3 |        2 | 
    2     |       76 |       20 |       11 |       85 |       23 |       13 | 
    3     |      680 |      172 |       86 |      765 |      195 |       99 | 
    4     |     5714 |     1432 |      722 |     6479 |     1627 |      821 | 
    5     |    47558 |    11900 |     5952 |    54037 |    13527 |     6773 | 
    6     |   376614 |    94183 |    47151 |   430651 |   107710 |    53924 | 
    7     |  2646584 |   661711 |   330973 |  3077235 |   769421 |   384897 | 
    8     | 13077539 |  3269570 |  1635502 | 16154774 |  4038991 |  2020399 | 
    9     | 23709256 |  5927616 |  2965278 | 39864030 |  9966607 |  4985677 | 
   10     |  5033865 |  1258677 |   630509 | 44897895 | 11225284 |  5616186 | 
   11     |     8505 |     2144 |     1120 | 44906400 | 11227428 |  5617306 | 

Monochrome upper face coset table for FTM:
Level     |          | SO       | GO       | cumul    | SO cumul | GO cumul |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |       12 |        3 |        2 |       13 |        4 |        3 | 
    2     |      150 |       39 |       21 |      163 |       43 |       24 | 
    3     |     1886 |      474 |      243 |     2049 |      517 |      267 | 
    4     |    21916 |     5486 |     2767 |    23965 |     6003 |     3034 | 
    5     |   242166 |    60565 |    30346 |   266131 |    66568 |    33380 | 
    6     |  2292695 |   573238 |   286803 |  2558826 |   639806 |   320183 | 
    7     | 14228012 |  3557182 |  1779298 | 16786838 |  4196988 |  2099481 | 
    8     | 25293406 |  6323720 |  3163613 | 42080244 | 10520708 |  5263094 | 
    9     |  2825994 |   706676 |   354185 | 44906238 | 11227384 |  5617279 | 
   10     |      162 |       44 |       27 | 44906400 | 11227428 |  5617306 | 

Monochrome upper face coset table for STM:
Level     |          | SO       | GO       | cumul    | SO cumul | GO cumul |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |       18 |        5 |        4 |       19 |        6 |        5 | 
    2     |      295 |       76 |       45 |      314 |       82 |       50 | 
    3     |     4654 |     1172 |      609 |     4968 |     1254 |      659 | 
    4     |    65864 |    16485 |     8298 |    70832 |    17739 |     8957 | 
    5     |   841872 |   210537 |   105458 |   912704 |   228276 |   114415 | 
    6     |  8130333 |  2032744 |  1016943 |  9043037 |  2261020 |  1131358 | 
    7     | 28593855 |  7148838 |  3575989 | 37636892 |  9409858 |  4707347 | 
    8     |  7266508 |  1816813 |   909558 | 44903400 | 11226671 |  5616905 | 
    9     |     3000 |      757 |      401 | 44906400 | 11227428 |  5617306 | 

Monochrome upper face coset table for ATM:
Level     |          | SO       | GO       | cumul    | SO cumul | GO cumul |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |       30 |        9 |        6 |       31 |       10 |        7 | 
    2     |      775 |      202 |      108 |      806 |      212 |      115 | 
    3     |    19350 |     4868 |     2468 |    20156 |     5080 |     2583 | 
    4     |   423558 |   105986 |    53112 |   443714 |   111066 |    55695 | 
    5     |  7513190 |  1878574 |   939752 |  7956904 |  1989640 |   995447 | 
    6     | 34069470 |  8517709 |  4260716 | 42026374 | 10507349 |  5256163 | 
    7     |  2880008 |   720074 |   361138 | 44906382 | 11227423 |  5617301 | 
    8     |       18 |        5 |        5 | 44906400 | 11227428 |  5617306 |
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Square group with center orientation

Submitted by brac37 on Sat, 10/23/2010 - 19:54.
Here is the square group with center orientation. 
Square group table with center orientation for FTM:
Level    |         | SO      | GO      | inv     | SO+inv  | GO+inv  |
---------+---------+---------+---------+---------+---------+---------+
    0    |       1 |       1 |       1 |       1 |       1 |       1 | 
    1    |       6 |       1 |       1 |       6 |       1 |       1 | 
    2    |      27 |       2 |       2 |      15 |       2 |       2 | 
    3    |     120 |       6 |       5 |      72 |       4 |       4 | 
    4    |     519 |      24 |      18 |     267 |      16 |      13 | 
    5    |    2088 |      93 |      59 |    1116 |      53 |      38 | 
    6    |    8368 |     360 |     210 |    4240 |     197 |     122 | 
    7    |   31470 |    1333 |     735 |   16026 |     691 |     403 | 
    8    |  110793 |    4655 |    2521 |   55773 |    2412 |    1345 | 
    9    |  348504 |   14605 |    7730 |  176334 |    7479 |    4094 | 
   10    |  937705 |   39215 |   20533 |  474389 |   20125 |   10724 | 
   11    | 1721148 |   71837 |   37138 |  870558 |   36759 |   19640 | 
   12    | 1532612 |   64096 |   33250 |  787392 |   33483 |   17710 | 
   13    |  541230 |   23115 |   12166 |  281274 |   12333 |    7056 | 
   14    |   62955 |    3297 |    1833 |   35543 |    2198 |    1239 | 
   15    |    9642 |     658 |     390 |    5526 |     376 |     252 | 
   16    |    1228 |     214 |     112 |    1228 |     214 |     112 | 

Square group table with center orientation for STM:
Level    |         | SO      | GO      | inv     | SO+inv  | GO+inv  |
---------+---------+---------+---------+---------+---------+---------+
    0    |       1 |       1 |       1 |       1 |       1 |       1 | 
    1    |       9 |       2 |       2 |       9 |       2 |       2 | 
    2    |      51 |       4 |       4 |      27 |       3 |       3 | 
    3    |     265 |      16 |      15 |     157 |      12 |      12 | 
    4    |    1290 |      63 |      51 |     678 |      39 |      33 | 
    5    |    5859 |     264 |     183 |    3039 |     147 |     110 | 
    6    |   25320 |    1103 |     680 |   12912 |     593 |     383 | 
    7    |   98934 |    4223 |    2405 |   50184 |    2205 |    1314 | 
    8    |  319074 |   13458 |    7282 |  161364 |    6950 |    3884 | 
    9    |  791788 |   33142 |   17405 |  399916 |   16959 |    9158 | 
   10    | 1430348 |   59687 |   30975 |  726036 |   30761 |   16412 | 
   11    | 1651969 |   68856 |   35284 |  835913 |   35240 |   18564 | 
   12    |  830644 |   34723 |   17989 |  433036 |   18473 |    9934 | 
   13    |  134144 |    6136 |    3440 |   73408 |    3470 |    2127 | 
   14    |   17352 |    1608 |     870 |   11760 |    1268 |     701 | 
   15    |    1356 |     223 |     116 |    1308 |     218 |     116 | 
   16    |      12 |       3 |       2 |      12 |       3 |       2 |
The center orientations on L, B, and D are determined by those on R, F and U, the corner coordinate and the edge coordinate. 
Square group table with center orientation for FTM, cumulative:
Level    |         | SO      | GO      | inv     | SO+inv  | GO+inv  |
---------+---------+---------+---------+---------+---------+---------+
    0    |       1 |       1 |       1 |       1 |       1 |       1 | 
    1    |       7 |       2 |       2 |       7 |       2 |       2 | 
    2    |      34 |       4 |       4 |      22 |       4 |       4 | 
    3    |     154 |      10 |       9 |      94 |       8 |       8 | 
    4    |     673 |      34 |      27 |     361 |      24 |      21 | 
    5    |    2761 |     127 |      86 |    1477 |      77 |      59 | 
    6    |   11129 |     487 |     296 |    5717 |     274 |     181 | 
    7    |   42599 |    1820 |    1031 |   21743 |     965 |     584 | 
    8    |  153392 |    6475 |    3552 |   77516 |    3377 |    1929 | 
    9    |  501896 |   21080 |   11282 |  253850 |   10856 |    6023 | 
   10    | 1439601 |   60295 |   31815 |  728239 |   30981 |   16747 | 
   11    | 3160749 |  132132 |   68953 | 1598797 |   67740 |   36387 | 
   12    | 4693361 |  196228 |  102203 | 2386189 |  101223 |   54097 | 
   13    | 5234591 |  219343 |  114369 | 2667463 |  113556 |   61153 | 
   14    | 5297546 |  222640 |  116202 | 2703006 |  115754 |   62392 | 
   15    | 5307188 |  223298 |  116592 | 2708532 |  116130 |   62644 | 
   16    | 5308416 |  223512 |  116704 | 2709760 |  116344 |   62756 | 

Square group table with center orientation for STM, cumulative:
Level    |         | SO      | GO      | inv     | SO+inv  | GO+inv  |
---------+---------+---------+---------+---------+---------+---------+
    0    |       1 |       1 |       1 |       1 |       1 |       1 | 
    1    |      10 |       3 |       3 |      10 |       3 |       3 | 
    2    |      61 |       7 |       7 |      37 |       6 |       6 | 
    3    |     326 |      23 |      22 |     194 |      18 |      18 | 
    4    |    1616 |      86 |      73 |     872 |      57 |      51 | 
    5    |    7475 |     350 |     256 |    3911 |     204 |     161 | 
    6    |   32795 |    1453 |     936 |   16823 |     797 |     544 | 
    7    |  131729 |    5676 |    3341 |   67007 |    3002 |    1858 | 
    8    |  450803 |   19134 |   10623 |  228371 |    9952 |    5742 | 
    9    | 1242591 |   52276 |   28028 |  628287 |   26911 |   14900 | 
   10    | 2672939 |  111963 |   59003 | 1354323 |   57672 |   31312 | 
   11    | 4324908 |  180819 |   94287 | 2190236 |   92912 |   49876 | 
   12    | 5155552 |  215542 |  112276 | 2623272 |  111385 |   59810 | 
   13    | 5289696 |  221678 |  115716 | 2696680 |  114855 |   61937 | 
   14    | 5307048 |  223286 |  116586 | 2708440 |  116123 |   62638 | 
   15    | 5308404 |  223509 |  116702 | 2709748 |  116341 |   62754 | 
   16    | 5308416 |  223512 |  116704 | 2709760 |  116344 |   62756 |
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two faces subgroup

Submitted by brac37 on Mon, 10/18/2010 - 20:26.
There was a bug in the program, but I think it is fixed now. The bug did not act on the square group. 
2faces group table for QTM:
Level     |          | SO       | GO       | inv      | SO+inv   | GO+inv   |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |        4 |        2 |        1 |        2 |        1 |        1 | 
    2     |       10 |        5 |        3 |        6 |        4 |        3 | 
    3     |       24 |       12 |        6 |       12 |        6 |        4 | 
    4     |       58 |       29 |       15 |       31 |       18 |       11 | 
    5     |      140 |       70 |       35 |       70 |       35 |       20 | 
    6     |      338 |      169 |       85 |      174 |       93 |       51 | 
    7     |      816 |      408 |      204 |      408 |      204 |      108 | 
    8     |     1970 |      985 |      493 |      997 |      513 |      267 | 
    9     |     4756 |     2378 |     1189 |     2378 |     1189 |      609 | 
   10     |    11448 |     5725 |     2863 |     5752 |     2916 |     1483 | 
   11     |    27448 |    13724 |     6862 |    13752 |     6876 |     3472 | 
   12     |    65260 |    32632 |    16324 |    32727 |    16461 |     8308 | 
   13     |   154192 |    77098 |    38550 |    77168 |    38585 |    19388 | 
   14     |   360692 |   180350 |    90192 |   180582 |    90522 |    45501 | 
   15     |   827540 |   413780 |   206898 |   413979 |   207074 |   103923 | 
   16     |  1851345 |   925690 |   462893 |   926328 |   463786 |   232692 | 
   17     |  3968840 |  1984439 |   992268 |  1985625 |   993015 |   497698 | 
   18     |  7891990 |  3946095 |  1973209 |  3947610 |  1975398 |   989918 | 
   19     | 13659821 |  6830037 |  3415314 |  6834924 |  3418469 |  1712711 | 
   20     | 18471682 |  9236067 |  4618491 |  9238500 |  4622960 |  2316366 | 
   21     | 16586822 |  8293763 |  4147448 |  8301511 |  4152530 |  2081704 | 
   22     |  8039455 |  4019999 |  2010449 |  4021563 |  2013156 |  1010156 | 
   23     |  1511110 |   755693 |   378110 |   757928 |   379817 |   191250 | 
   24     |    47351 |    23701 |    11894 |    23737 |    12005 |     6152 | 
   25     |       87 |       44 |       27 |       53 |       32 |       21 | 

2faces group table for FTM:
Level     |          | SO       | GO       | inv      | SO+inv   | GO+inv   |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |        6 |        3 |        2 |        4 |        2 |        2 | 
    2     |       18 |        9 |        5 |        9 |        6 |        4 | 
    3     |       54 |       27 |       14 |       30 |       15 |       10 | 
    4     |      162 |       81 |       41 |       81 |       45 |       25 | 
    5     |      486 |      243 |      122 |      252 |      126 |       70 | 
    6     |     1457 |      729 |      365 |      729 |      378 |      196 | 
    7     |     4360 |     2180 |     1091 |     2206 |     1103 |      572 | 
    8     |    13016 |     6508 |     3256 |     6528 |     3303 |     1672 | 
    9     |    38482 |    19243 |     9627 |    19345 |     9692 |     4911 | 
   10     |   113094 |    56548 |    28282 |    56605 |    28441 |    14301 | 
   11     |   328920 |   164462 |    82243 |   164864 |    82440 |    41454 | 
   12     |   942351 |   471183 |   235611 |   471307 |   236060 |   118393 | 
   13     |  2616973 |  1308507 |   654297 |  1309754 |   655082 |   328547 | 
   14     |  6774848 |  3387472 |  1693858 |  3388767 |  1695717 |   849714 | 
   15     | 15105592 |  7552965 |  3776718 |  7557196 |  3780187 |  1894131 | 
   16     | 24231019 | 12115830 |  6058483 | 12121324 |  6064602 |  3038464 | 
   17     | 19421274 |  9711098 |  4856334 |  9718187 |  4862261 |  2437950 | 
   18     |  3843568 |  1922014 |   961504 |  1924617 |   964036 |   485064 | 
   19     |    47465 |    23763 |    11954 |    23984 |    12152 |     6325 | 
   20     |       54 |       30 |       16 |       28 |       17 |       12 |
Here are the cumulative results. They are computed separately, which is not very smart, but it is only a small group. 
2faces group table for QTM, cumulative:
Level     |          | SO       | GO       | inv      | SO+inv   | GO+inv   |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |        5 |        3 |        2 |        3 |        2 |        2 | 
    2     |       15 |        8 |        5 |        9 |        6 |        5 | 
    3     |       39 |       20 |       11 |       21 |       12 |        9 | 
    4     |       97 |       49 |       26 |       52 |       30 |       20 | 
    5     |      237 |      119 |       61 |      122 |       65 |       40 | 
    6     |      575 |      288 |      146 |      296 |      158 |       91 | 
    7     |     1391 |      696 |      350 |      704 |      362 |      199 | 
    8     |     3361 |     1681 |      843 |     1701 |      875 |      466 | 
    9     |     8117 |     4059 |     2032 |     4079 |     2064 |     1075 | 
   10     |    19565 |     9784 |     4895 |     9831 |     4980 |     2558 | 
   11     |    47013 |    23508 |    11757 |    23583 |    11856 |     6030 | 
   12     |   112273 |    56140 |    28081 |    56310 |    28317 |    14338 | 
   13     |   266465 |   133238 |    66631 |   133478 |    66902 |    33726 | 
   14     |   627157 |   313588 |   156823 |   314060 |   157424 |    79227 | 
   15     |  1454697 |   727368 |   363721 |   728039 |   364498 |   183150 | 
   16     |  3306042 |  1653058 |   826614 |  1654367 |   828284 |   415842 | 
   17     |  7274882 |  3637497 |  1818882 |  3639992 |  1821299 |   913540 | 
   18     | 15166872 |  7583592 |  3792091 |  7587602 |  3796697 |  1903458 | 
   19     | 28826693 | 14413629 |  7207405 | 14422526 |  7215166 |  3616169 | 
   20     | 47298375 | 23649696 | 11825896 | 23661026 | 11838126 |  5932535 | 
   21     | 63885197 | 31943459 | 15973344 | 31962537 | 15990656 |  8014239 | 
   22     | 71924652 | 35963458 | 17983793 | 35984100 | 18003812 |  9024395 | 
   23     | 73435762 | 36719151 | 18361903 | 36742028 | 18383629 |  9215645 | 
   24     | 73483113 | 36742852 | 18373797 | 36765765 | 18395634 |  9221797 | 
   25     | 73483200 | 36742896 | 18373824 | 36765818 | 18395666 |  9221818 | 

2faces group table for FTM, cumulative:
Level     |          | SO       | GO       | inv      | SO+inv   | GO+inv   |
----------+----------+----------+----------+----------+----------+----------+
    0     |        1 |        1 |        1 |        1 |        1 |        1 | 
    1     |        7 |        4 |        3 |        5 |        3 |        3 | 
    2     |       25 |       13 |        8 |       14 |        9 |        7 | 
    3     |       79 |       40 |       22 |       44 |       24 |       17 | 
    4     |      241 |      121 |       63 |      125 |       69 |       42 | 
    5     |      727 |      364 |      185 |      377 |      195 |      112 | 
    6     |     2184 |     1093 |      550 |     1106 |      573 |      308 | 
    7     |     6544 |     3273 |     1641 |     3312 |     1676 |      880 | 
    8     |    19560 |     9781 |     4897 |     9840 |     4979 |     2552 | 
    9     |    58042 |    29024 |    14524 |    29185 |    14671 |     7463 | 
   10     |   171136 |    85572 |    42806 |    85790 |    43112 |    21764 | 
   11     |   500056 |   250034 |   125049 |   250654 |   125552 |    63218 | 
   12     |  1442407 |   721217 |   360660 |   721961 |   361612 |   181611 | 
   13     |  4059380 |  2029724 |  1014957 |  2031715 |  1016694 |   510158 | 
   14     | 10834228 |  5417196 |  2708815 |  5420482 |  2712411 |  1359872 | 
   15     | 25939820 | 12970161 |  6485533 | 12977678 |  6492598 |  3254003 | 
   16     | 50170839 | 25085991 | 12544016 | 25099002 | 12557200 |  6292467 | 
   17     | 69592113 | 34797089 | 17400350 | 34817189 | 17419461 |  8730417 | 
   18     | 73435681 | 36719103 | 18361854 | 36741806 | 18383497 |  9215481 | 
   19     | 73483146 | 36742866 | 18373808 | 36765790 | 18395649 |  9221806 | 
   20     | 73483200 | 36742896 | 18373824 | 36765818 | 18395666 |  9221818 |
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Excuse my ignorance possibly

Submitted by secondmouse on Tue, 10/19/2010 - 15:01.
Does this mean the diameter of the Cayley graph of the two-face group in the QTM is 25? And that there are 87 antipodes?
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Indeed

Submitted by brac37 on Tue, 10/19/2010 - 18:50.
87 - 53 = 34 of them are selfinverse, and there are only 21 "really different" antipodes. One can choose the first turn in four ways and every next turn in only two and a half ways, which explains why this small group still has a large diameter, more or less the same diameter as the whole cube.
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Cayley graph of the subgroup

Submitted by mdlazreg on Mon, 10/25/2010 - 18:48.
What about the three-face subgroup? The one generated by U,R,F? Is your program able to generate its QTM distance table?


I find it amazing that the U,R subgroup has a depth of 25! not very far from the Rubik's God number! and I wonder what God's number for U,R,F is...
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Indeed - me too...

Submitted by secondmouse on Tue, 10/26/2010 - 04:48.
I suppose that may be too much to hope for at this juncture given the size of the group (17065973514200 elements) and the amount of memory it must take (although must admit partial ignorance of the methods used).

As you've probably seen B MacKenzie has compiled a complete table of the corners only and edges only positions in the QTM and has combined these to list the number of elements out to length 18 in the 3-face subgroup (at least that's my reading of it).

To be frank all I'm really interested in is the number of actual positions of a given depth or length. It would be nice anyway to have independent verification out to depth 18 in the QTM for the 3-face subgroup.
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U,R,F might be reachable

Submitted by brac37 on Tue, 10/26/2010 - 17:40.
The project is not yet developed to do large stuff. The following things are still to be added:
  • Symmetry reduction. Currently, a depth table for all positions is generated. Algebra with (anti)symmetries is already implemented, though (otherwise the (anti)symmetry columns could not be computed efficiently: the program takes the intersection of (anti)symmetry subgroups of the coordinates and counts the position 1/s times, s being the size of the intersection).
  • Quaternary table. Currently, one byte is used for each position. A quaternary table is too small for a depth table, but large enough to generate the above position count tables in the same time as with a depth table. The quaternary table codes unreached, new (to be reached), just reached, old (does not have new neighbors).
  • Harddisk memory. Currently, all positions are stored in RAM. This is not necessary. One can divide the positions in e.g. edge (position) cosets and do (anti)symmetry reduction on the edge (position). Now for every edge (position) coset, the number of neighboring cosets is equal to the number of primitive face turns (in case of symmetry reduction), or the double of this number (in case of antisymmetry reduction). Thus to update an edge (position) coset one level, a few other edge (position) cosets need to be read (every coset needs to be written twice and read 2*6+2=14 times for each level with QTM and 2*9+2=20 times for each level with FTM).
With antisymmetry reduction, 17065973514200/3!/2 position remain. Next, four positions go into a byte such that 3555411148800 bytes are necessary. That is with perfect symmetry reduction, but 4TBytes of harddisk space will be enough. 4TBytes of harddisk space can be arranged. I do not think that time will be a real problem either: I guess it will take less than a year with not to many cores and a fast RAID cluster.

Note that the positions generated by U,R,F are solved with U,R,F. Solving them within the whole cube requires a coset solver, which B MacKenzie has, but time matters for such a solver. There might be possible improvements for MacKenzie's solver though, such as not going beyond level 24 with QTM. Level 25 and up is about 0.1 percent, of which most are odd by far. All neighbors of even > 24 positions (which are odd > 24 positions) are to be solved with a fast optimal solver, I guess. That will complete the calculation.

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