The first table shows all symmetric positions, counts and
depths. These numbers all agree with Radu and Kociemba.
Summary in HTM
dist pos modM modM+inv mod48
---- --------------- ------------- ------------- -----
0 1 1 1 1
1 18 2 2 18
2 51 5 5 3
3 312 14 8 24
4 1,335 61 42 39
5 4,380 191 115 12
6 17,782 784 461 22
7 70,188 2,974 1,631 12
8 229,336 9,729 5,294 40
9 851,139 35,615 18,604 3
10 2,989,204 125,103 65,125 4
11 9,732,164 406,615 208,603 20
12 35,024,904 1,461,865 746,313 24
13 122,054,340 5,089,766 2,581,241 36
14 436,197,214 18,183,156 9,195,333 46
15 1,763,452,505 73,491,944 37,037,332 41
16 8,035,307,127 334,845,106 168,373,488 39
17 37,542,012,922 1,564,371,587 785,249,161 10
18 95,387,902,305 3,974,821,633 1,994,938,045 33
19 21,267,102,443 886,299,819 446,607,275 11
20 1,091,994 46,514 32,625 42
---- --------------- ------------- ------------- -----
tot 164,604,041,664 6,859,192,484 3,445,060,704 0
The remaining tables give counts for all the individual symmetry
subgroups. In the tables below, the "pos" column gives all
positions in the subgroup (including those that happen to have
more symmetry); the "modM" and "modM+inv" columns however are
for only positions with *exactly* that symmetry.
Summary of C_2_(a) in HTM
dist pos modM modM+inv
---- -------------- ------------- -----------
0 1 0 0
1 6 0 0
2 15 0 0
3 72 8 4
4 343 34 22
5 1,060 123 69
6 4,570 507 277
7 17,960 2,182 1,143
8 60,136 7,210 3,793
9 226,199 28,017 14,361
10 805,260 99,641 50,974
11 2,602,404 323,655 164,112
12 9,295,556 1,158,307 585,163
13 30,949,920 3,863,416 1,945,189
14 101,692,042 12,702,003 6,387,506
15 363,402,817 45,411,240 22,803,625
16 1,333,918,995 166,705,822 83,639,845
17 4,550,677,966 568,759,399 285,178,698
18 7,774,262,733 971,648,345 487,396,809
19 1,120,268,931 139,966,952 70,596,762
20 51,094 5,763 4,464
---- -------------- ------------- -----------
tot 15,288,238,080 1,910,682,624 958,772,816
Summary of C_2_(b) in HTM
dist pos modM modM+inv
---- ------------- ----------- -----------
0 1 0 0
2 3 0 0
4 1 0 0
6 44 2 1
7 180 33 23
8 460 78 49
9 1,587 365 225
10 8,072 1,861 1,122
11 28,146 6,713 3,826
12 119,054 29,113 16,120
13 473,702 117,581 63,201
14 2,168,270 540,239 282,525
15 12,440,909 3,107,127 1,592,911
16 84,769,773 21,185,220 10,730,009
17 548,185,762 137,030,323 69,003,014
18 1,583,371,387 395,812,880 199,227,132
19 316,458,701 79,102,176 40,162,141
20 13,628 3,297 2,725
---- ------------- ----------- -----------
tot 2,548,039,680 636,937,008 321,085,024
Summary of C_2h_(a) in HTM
dist pos modM modM+inv
---- ------- ------- --------
0 1 0 0
2 5 0 0
4 23 2 1
5 8 2 1
6 102 5 4
7 34 7 7
8 322 42 35
9 137 33 29
10 500 115 89
11 1,768 423 327
12 2,916 622 521
13 6,002 1,285 1,043
14 12,426 2,489 1,997
15 19,795 4,636 3,566
16 46,029 11,153 8,057
17 108,816 26,636 17,388
18 239,765 58,874 35,586
19 149,539 36,878 21,920
20 1,636 350 293
---- ------- ------- --------
tot 589,824 143,552 90,864
Summary of C_2h_(b) in HTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
2 1 0 0
4 1 0 0
6 6 1 1
7 6 1 1
8 28 7 7
9 23 10 10
10 90 33 33
11 200 72 72
12 474 192 191
13 786 329 323
14 1,470 671 634
15 3,585 1,720 1,471
16 8,951 4,360 3,496
17 21,560 10,513 7,953
18 42,015 20,776 15,207
19 18,953 9,360 7,384
20 154 71 69
---- ------ ------ --------
tot 98,304 48,116 36,852
Summary of C_2v_(a1) in HTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
1 2 0 0
2 1 0 0
4 1 0 0
5 4 0 0
6 4 0 0
7 10 1 1
8 26 1 1
9 9 1 1
10 146 22 20
11 462 92 73
12 756 134 107
13 832 130 102
14 3,112 704 554
15 5,249 1,249 993
16 9,769 2,347 1,824
17 15,690 3,735 2,933
18 22,009 5,340 4,174
19 7,285 1,757 1,530
20 168 39 39
---- ------ ------ --------
tot 65,536 15,552 12,352
Summary of C_2v_(a2) in HTM
dist pos modM modM+inv
---- --------- ------- --------
0 1 0 0
1 2 0 0
2 3 0 0
3 8 2 1
4 15 0 0
5 36 8 6
6 92 4 3
7 140 33 21
8 230 17 13
9 167 34 26
10 494 111 86
11 1,184 280 203
12 4,284 960 715
13 9,252 2,080 1,551
14 20,858 4,549 3,313
15 32,193 7,712 5,542
16 89,617 22,012 14,838
17 223,444 55,305 34,551
18 505,801 125,363 72,962
19 289,047 71,764 41,100
20 2,780 646 445
---- --------- ------- --------
tot 1,179,648 290,880 175,376
Summary of C_2v_(b) in HTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
2 1 0 0
4 1 0 0
6 10 3 3
7 6 1 1
8 20 3 3
9 39 18 18
10 92 34 34
11 172 58 58
12 456 183 180
13 714 293 289
14 1,468 671 622
15 3,127 1,491 1,292
16 8,993 4,381 3,625
17 21,186 10,329 7,840
18 42,201 20,873 15,168
19 19,617 9,696 7,638
20 200 94 93
---- ------ ------ --------
tot 98,304 48,128 36,864
Summary of C_3 in HTM
dist pos modM modM+inv
---- --------- ------- --------
0 1 0 0
6 1 0 0
8 4 0 0
9 18 3 3
10 28 6 5
11 26 5 3
12 162 32 28
13 288 71 46
14 1,711 418 260
15 8,924 2,216 1,238
16 67,176 16,705 9,105
17 481,063 120,050 62,806
18 2,083,458 520,076 271,003
19 1,133,447 282,475 152,809
20 2,829 659 576
---- --------- ------- --------
tot 3,779,136 942,716 497,882
Summary of C_3v in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
6 1 0 0
13 4 2 2
14 3 1 1
15 6 3 3
16 4 2 2
17 7 2 2
18 12 4 4
19 9 2 2
20 1 0 0
---- --- ---- --------
tot 48 16 16
Summary of C_4 in HTM
dist pos modM modM+inv
---- ------- ------ --------
0 1 0 0
1 6 1 1
2 9 1 1
4 1 0 0
5 12 2 1
6 50 8 5
7 42 7 5
8 56 7 5
9 33 7 7
10 188 42 41
11 758 177 131
12 1,720 381 267
13 2,604 583 389
14 4,040 958 674
15 9,401 2,302 1,507
16 21,803 5,362 3,148
17 46,296 11,461 6,475
18 48,801 12,066 7,118
19 11,413 2,756 1,852
20 222 39 37
---- ------- ------ --------
tot 147,456 36,160 21,664
Summary of C_4h in HTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
2 3 1 1
4 1 0 0
6 10 3 3
7 2 1 1
8 12 3 3
9 1 0 0
10 2 0 0
11 20 10 10
12 62 28 27
13 84 35 35
14 58 20 15
15 71 34 29
16 193 87 76
17 210 95 86
18 387 184 162
19 353 171 132
20 66 32 28
---- ----- ---- --------
tot 1,536 704 608
Summary of C_4v in HTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
1 2 1 1
2 1 0 0
4 1 0 0
5 4 2 1
6 4 0 0
7 2 1 1
8 12 3 2
9 3 1 1
10 14 6 6
11 20 10 10
12 80 37 32
13 154 70 57
14 152 67 63
15 101 49 39
16 141 61 57
17 186 83 75
18 125 53 51
19 19 4 4
20 2 0 0
---- ----- ---- --------
tot 1,024 448 400
Summary of C_i in HTM
dist pos modM modM+inv
---- -------------- ------------- -----------
0 1 0 0
2 9 0 0
4 63 1 1
5 120 4 4
6 694 22 18
7 2,562 102 66
8 11,338 440 262
9 44,853 1,846 998
10 194,740 8,036 4,360
11 788,992 32,617 16,884
12 3,315,172 137,692 71,004
13 13,445,694 559,379 284,571
14 54,961,138 2,288,348 1,160,146
15 271,122,047 11,293,497 5,689,550
16 1,585,605,201 66,058,983 33,189,184
17 9,134,397,172 380,581,022 190,837,230
18 27,801,243,853 1,158,344,598 580,705,104
19 6,999,321,491 291,614,579 146,586,998
20 259,100 10,540 7,188
---- -------------- ------------- -----------
tot 45,864,714,240 1,910,931,706 958,553,568
Summary of C_s_(a) in HTM
dist pos modM modM+inv
---- -------------- ------------- -------------
0 1 0 0
1 4 0 0
2 13 1 1
3 48 4 3
4 159 17 12
5 464 48 32
6 1,554 176 104
7 4,758 555 324
8 13,810 1,662 931
9 40,353 4,977 2,719
10 114,084 14,062 7,601
11 324,496 40,016 21,300
12 1,019,512 125,984 66,943
13 4,128,898 513,072 266,473
14 20,060,486 2,501,038 1,280,752
15 103,107,447 12,877,423 6,527,551
16 607,357,293 75,889,571 38,240,126
17 3,601,456,840 450,107,871 226,052,001
18 11,082,306,785 1,385,122,357 695,129,013
19 2,925,751,099 365,623,453 184,049,962
20 197,592 23,793 15,592
---- -------------- ------------- -------------
tot 18,345,885,696 2,292,846,080 1,151,661,440
Summary of C_s_(b) in HTM
dist pos modM modM+inv
---- ----------- ----------- ----------
0 1 0 0
1 2 0 0
2 1 0 0
4 1 0 0
5 4 0 0
6 24 3 2
7 50 9 5
8 126 20 12
9 533 117 83
10 1,956 419 267
11 6,478 1,439 883
12 25,130 5,906 3,422
13 114,700 28,154 15,734
14 546,558 135,189 72,285
15 3,102,247 772,641 402,389
16 19,703,503 4,919,061 2,517,679
17 110,503,448 27,611,515 14,012,771
18 252,385,203 63,069,968 32,023,935
19 38,279,025 9,558,403 4,965,270
20 4,290 948 863
---- ----------- ----------- ----------
tot 424,673,280 106,103,792 54,015,600
Summary of D_2_(edge) in HTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
2 3 0 0
4 1 0 0
6 28 4 4
7 44 6 6
8 128 22 18
9 71 10 9
10 494 111 91
11 1,030 230 172
12 1,844 403 304
13 2,134 461 373
14 4,604 1,064 861
15 5,959 1,396 1,111
16 11,355 2,667 2,051
17 22,676 5,397 3,980
18 36,221 8,736 5,922
19 11,617 2,713 1,807
20 94 12 11
---- ------ ------ --------
tot 98,304 23,232 16,720
Summary of D_2_(face) in HTM
dist pos modM modM+inv
---- ------- ------ --------
0 1 0 0
2 9 0 0
4 63 4 3
5 24 2 1
6 424 27 18
7 246 18 10
8 2,000 152 98
9 1,593 132 79
10 6,282 515 321
11 7,776 641 387
12 17,780 1,428 881
13 20,250 1,608 897
14 28,698 2,185 1,257
15 23,343 1,850 1,135
16 45,731 3,708 2,207
17 66,418 5,370 3,104
18 57,353 4,456 2,601
19 16,711 1,256 749
20 210 4 4
---- ------- ------ --------
tot 294,912 23,356 13,752
Summary of D_2d_(edge) in HTM
dist pos modM modM+inv
---- ----- ----- --------
0 1 0 0
2 1 0 0
4 1 0 0
6 4 0 0
7 6 3 3
8 22 8 8
9 27 13 11
10 22 10 6
11 44 22 18
12 88 41 39
13 128 57 52
14 224 103 98
15 209 103 93
16 435 208 161
17 504 242 192
18 797 389 288
19 523 256 172
20 36 17 11
---- ----- ----- --------
tot 3,072 1,472 1,152
Summary of D_2d_(face) in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
2 1 0 0
4 1 0 0
6 4 0 0
8 8 1 1
9 1 0 0
10 24 11 9
11 18 9 9
12 62 28 27
13 58 22 19
14 54 18 16
15 13 5 5
16 47 14 12
17 76 28 24
18 107 44 42
19 35 12 12
20 2 0 0
---- --- ---- --------
tot 512 192 176
Summary of D_2h_(edge) in HTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
2 1 0 0
4 1 0 0
6 4 0 0
7 4 2 2
8 14 4 4
9 3 1 1
10 24 11 11
11 56 28 28
12 90 42 42
13 128 57 57
14 126 54 54
15 145 71 71
16 231 106 106
17 528 254 254
18 455 218 218
19 225 107 107
20 12 5 5
---- ----- ---- --------
tot 2,048 960 960
Summary of D_2h_(face) in HTM
dist pos modM modM+inv
---- ------ ----- --------
0 1 0 0
2 3 0 0
4 15 2 2
6 76 11 11
8 140 20 20
9 3 0 0
10 18 2 2
12 288 45 45
13 678 106 106
14 2,322 378 375
15 1,041 172 164
16 1,031 161 137
17 1,574 252 244
18 3,465 568 406
19 1,471 240 143
20 162 25 17
---- ------ ----- --------
tot 12,288 1,982 1,672
Summary of D_3 in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
6 1 0 0
11 2 1 1
12 4 2 2
14 13 6 5
15 10 5 5
16 28 14 11
17 55 26 18
18 174 85 62
19 135 65 52
20 9 4 4
---- --- ---- --------
tot 432 208 160
Summary of D_3d in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
6 1 0 0
14 1 1 1
17 3 3 3
18 4 4 4
19 5 4 4
20 1 0 0
---- --- ---- --------
tot 16 12 12
Summary of D_4 in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
2 3 1 1
4 1 0 0
6 12 4 3
7 10 5 4
8 16 5 5
9 3 1 1
10 8 3 3
11 10 5 5
12 66 30 27
13 62 24 21
14 34 8 8
15 27 12 12
16 59 20 16
17 96 38 34
18 63 22 22
19 39 14 14
20 2 0 0
---- --- ---- --------
tot 512 192 176
Summary of D_4h in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
2 1 1 1
4 1 1 1
6 4 3 3
8 6 6 6
9 1 1 1
10 2 2 2
12 6 6 6
13 14 14 14
14 18 18 18
15 3 3 3
16 19 19 19
17 20 20 20
18 19 19 19
19 11 10 10
20 2 1 1
---- --- ---- --------
tot 128 124 124
Summary of O_h in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 1 1
6 1 1 1
19 1 1 1
20 1 1 1
---- --- ---- --------
tot 4 4 4
Summary of S_4 in HTM
dist pos modM modM+inv
---- ------- ------- --------
0 1 0 0
2 3 0 0
4 1 0 0
6 10 0 0
7 36 7 7
8 94 16 16
9 127 25 19
10 236 48 40
11 522 110 90
12 1,224 256 210
13 3,990 937 697
14 8,076 1,944 1,283
15 20,147 4,965 3,011
16 52,237 12,900 7,588
17 125,622 31,218 17,201
18 176,669 43,854 24,049
19 52,945 13,014 7,554
20 428 82 75
---- ------- ------- --------
tot 442,368 109,376 61,840
Summary of S_6 in HTM
dist pos modM modM+inv
---- ----- ----- --------
0 1 0 0
6 1 0 0
8 4 1 1
9 6 3 2
10 4 2 2
11 4 2 1
12 22 11 8
14 25 12 10
15 44 22 16
16 324 158 106
17 799 397 261
18 2,956 1,476 979
19 3,401 1,698 1,143
20 185 88 81
---- ----- ----- --------
tot 7,776 3,870 2,610
Summary of T in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
6 1 0 0
8 2 0 0
12 8 2 2
16 8 0 0
17 10 2 2
18 20 5 5
19 13 3 3
20 9 0 0
---- --- ---- --------
tot 72 12 12
Summary of T_h in HTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
6 1 0 0
8 2 1 1
16 8 4 3
17 2 1 1
19 1 0 0
20 9 4 3
---- --- ---- --------
tot 24 10 8
Summary of hard positions in HTM
symm pos modM modM+inv
----------- --------- ------ --------
O_h 1 1 1
D_4h 3 1 1
T_h 8 4 3
D_2h_(edge) 30 5 5
D_3 32 4 4
D_2_(face) 48 4 4
D_2d_(edge) 102 17 11
D_2_(edge) 144 12 11
D_2h_(face) 150 25 17
C_4h 192 32 28
C_2v_(a1) 468 39 39
C_4 468 39 37
S_6 704 88 81
C_2h_(b) 852 71 69
S_4 984 82 75
C_2v_(b) 1,128 94 93
C_2h_(a) 4,200 350 293
C_2v_(a2) 7,752 646 445
C_3 10,544 659 576
C_s_(b) 22,752 948 863
C_2_(b) 79,128 3,297 2,725
C_2_(a) 138,312 5,763 4,464
C_i 252,960 10,540 7,188
C_s_(a) 571,032 23,793 15,592
----------- --------- ------ --------
tot 1,091,994 46,514 32,625