The first table shows all symmetric positions, counts and
depths. These are new results as of March 2014.
Summary of C_1 in QTM
dist pos modM modM+inv mod48
---- --------------- ------------- ------------- -----
0 1 1 1 1
1 12 1 1 12
2 18 3 3 18
3 108 5 3 12
4 411 19 14 27
5 1,104 46 25 0
6 3,744 163 102 0
7 11,760 490 258 0
8 36,731 1,582 902 11
9 111,144 4,632 2,370 24
10 358,138 15,054 7,908 10
11 1,028,848 42,895 21,647 16
12 3,266,949 136,691 70,284 21
13 9,443,588 393,602 198,149 20
14 29,201,318 1,218,544 618,945 38
15 83,765,676 3,490,523 1,752,632 12
16 268,136,523 11,177,948 5,644,771 27
17 819,440,112 34,146,994 17,140,681 0
18 3,083,699,868 128,516,587 64,687,074 12
19 11,628,867,276 484,563,973 243,091,188 12
20 41,538,350,563 1,730,948,894 869,224,590 19
21 67,617,360,740 2,817,563,628 1,413,565,363 20
22 37,373,063,137 1,557,455,774 783,700,221 1
23 2,147,815,036 89,510,797 45,330,245 28
24 78,820 3,635 3,324 4
25 36 2 2 36
26 3 1 1 3
---- --------------- ------------- ------------- -----
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 QTM
dist pos modM modM+inv
---- -------------- ------------- -----------
0 1 0 0
1 4 0 0
2 10 0 0
3 36 4 2
4 123 12 8
5 368 46 25
6 1,192 132 78
7 3,856 482 254
8 11,411 1,345 724
9 36,296 4,536 2,318
10 112,410 13,792 7,114
11 335,920 41,965 21,157
12 1,020,073 126,636 64,312
13 3,058,828 382,235 192,220
14 8,978,902 1,119,811 564,679
15 26,377,392 3,296,942 1,653,623
16 77,108,783 9,631,733 4,840,286
17 227,640,076 28,452,180 14,264,211
18 685,484,820 85,655,163 42,991,338
19 2,007,223,500 250,890,643 125,744,086
20 4,656,375,131 581,918,540 291,904,955
21 5,304,119,504 662,991,213 332,570,156
22 2,215,021,221 276,741,182 139,150,287
23 75,323,248 9,413,673 4,800,651
24 4,960 358 331
25 12 1 1
26 3 0 0
---- -------------- ------------- -----------
tot 15,288,238,080 1,910,682,624 958,772,816
Summary of C_2_(b) in QTM
dist pos modM modM+inv
---- ------------- ----------- -----------
0 1 0 0
2 2 0 0
4 1 0 0
7 32 8 4
8 181 30 21
9 312 78 41
10 898 188 113
11 2,560 640 341
12 7,945 1,814 1,088
13 29,548 7,387 3,823
14 91,464 22,462 12,307
15 385,910 96,454 49,328
16 1,537,621 383,032 198,706
17 7,225,614 1,806,031 914,400
18 36,699,180 9,169,755 4,651,890
19 180,982,220 45,241,940 22,772,973
20 681,459,499 170,342,729 85,808,746
21 1,061,220,706 265,286,959 133,536,783
22 554,222,051 138,536,519 70,017,794
23 24,172,938 6,040,783 3,116,475
24 996 199 191
26 1 0 0
---- ------------- ----------- -----------
tot 2,548,039,680 636,937,008 321,085,024
Summary of C_2h_(a) in QTM
dist pos modM modM+inv
---- ------- ------- --------
0 1 0 0
2 2 0 0
4 3 0 0
6 8 2 1
8 79 10 9
10 96 22 19
12 759 135 105
14 1,712 364 304
16 8,013 1,755 1,321
18 34,418 7,893 5,624
20 229,733 55,892 35,839
22 314,539 77,374 47,539
24 458 105 103
26 3 0 0
---- ------- ------- --------
tot 589,824 143,552 90,864
Summary of C_2h_(b) in QTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
4 1 0 0
8 9 4 4
10 8 3 3
12 107 38 38
14 310 121 121
15 50 25 20
16 1,169 495 484
17 828 413 278
18 4,444 2,030 1,899
19 7,896 3,947 2,633
20 20,703 10,013 8,380
21 35,616 17,802 12,031
22 22,347 10,823 9,014
23 4,762 2,381 1,926
24 52 21 21
26 1 0 0
---- ------ ------ --------
tot 98,304 48,116 36,852
Summary of C_2v_(a1) in QTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
2 2 0 0
4 1 0 0
8 1 0 0
10 8 0 0
12 97 8 6
14 448 66 56
15 156 39 27
16 1,179 227 173
17 988 247 183
18 5,222 1,119 877
19 4,744 1,186 908
20 17,395 4,060 3,212
21 22,660 5,665 4,318
22 8,369 1,873 1,621
23 4,220 1,055 964
24 44 7 7
26 1 0 0
---- ------ ------ --------
tot 65,536 15,552 12,352
Summary of C_2v_(a2) in QTM
dist pos modM modM+inv
---- --------- ------- --------
0 1 0 0
2 2 0 0
4 3 0 0
6 8 2 1
8 71 6 3
10 108 23 16
12 691 105 78
14 2,622 587 415
16 10,627 2,411 1,737
18 60,046 14,253 9,436
20 461,903 113,824 68,566
22 642,105 159,312 94,815
24 1,458 357 309
26 3 0 0
---- --------- ------- --------
tot 1,179,648 290,880 175,376
Summary of C_2v_(b) in QTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
4 1 0 0
8 5 2 2
10 12 5 5
12 81 25 25
14 208 70 70
15 44 22 18
16 1,049 435 425
17 662 331 231
18 3,980 1,799 1,716
19 6,504 3,252 2,250
20 19,581 9,454 8,010
21 37,002 18,501 12,423
22 24,163 11,732 9,686
23 4,940 2,470 1,974
24 70 30 29
26 1 0 0
---- ------ ------ --------
tot 98,304 48,128 36,864
Summary of C_3 in QTM
dist pos modM modM+inv
---- --------- ------- --------
0 1 0 0
8 2 0 0
10 4 1 1
11 16 4 2
12 51 9 7
13 8 2 1
14 254 60 41
15 360 89 48
16 2,004 487 309
17 5,397 1,345 705
18 31,131 7,732 4,431
19 142,371 35,515 18,206
20 678,691 169,386 90,740
21 1,572,830 392,559 199,807
22 1,176,979 293,578 161,531
23 168,586 41,850 21,959
24 451 99 94
---- --------- ------- --------
tot 3,779,136 942,716 497,882
Summary of C_3v in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
12 1 0 0
16 2 1 1
17 3 1 1
18 5 2 2
19 7 3 3
20 7 2 2
21 10 2 2
22 7 3 3
23 4 2 2
24 1 0 0
---- --- ---- --------
tot 48 16 16
Summary of C_4 in QTM
dist pos modM modM+inv
---- ------- ------ --------
0 1 0 0
1 4 1 1
2 6 0 0
3 4 1 1
4 1 0 0
8 1 0 0
9 8 2 1
10 42 6 5
11 160 40 21
12 407 85 50
13 820 205 125
14 972 199 148
15 912 228 146
16 2,095 486 340
17 8,252 2,063 1,225
18 18,524 4,505 2,727
19 26,848 6,712 3,817
20 33,399 8,092 4,920
21 34,476 8,619 4,868
22 18,205 4,343 2,838
23 2,240 560 418
24 74 12 12
25 4 1 1
26 1 0 0
---- ------- ------ --------
tot 147,456 36,160 21,664
Summary of C_4h in QTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
2 2 1 1
4 1 0 0
8 1 0 0
10 6 3 3
12 15 1 1
14 76 33 33
16 51 17 17
18 172 78 77
20 529 246 217
22 659 315 250
24 22 10 9
26 1 0 0
---- ----- ---- --------
tot 1,536 704 608
Summary of C_4v in QTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
2 2 1 1
4 1 0 0
8 1 0 0
10 4 2 1
12 35 11 10
14 70 30 26
16 71 27 25
18 286 135 114
20 399 181 163
22 149 60 59
24 4 1 1
26 1 0 0
---- ----- ---- --------
tot 1,024 448 400
Summary of C_i in QTM
dist pos modM modM+inv
---- -------------- ------------- -----------
0 1 0 0
2 6 0 0
4 27 1 1
6 120 4 4
8 863 27 23
9 192 8 4
10 7,620 304 182
11 2,208 92 47
12 75,567 3,051 1,720
13 34,344 1,431 737
14 739,556 30,549 16,360
15 947,548 39,468 20,036
16 12,058,761 501,258 258,311
17 40,061,232 1,669,009 839,237
18 359,580,434 14,977,366 7,553,717
19 2,012,791,496 83,864,293 42,035,816
20 10,624,291,329 442,645,324 222,062,307
21 20,173,117,696 840,537,259 421,158,871
22 11,935,588,411 497,271,348 249,789,819
23 705,402,404 29,390,388 14,815,911
24 14,422 526 465
26 3 0 0
---- -------------- ------------- -----------
tot 45,864,714,240 1,910,931,706 958,553,568
Summary of C_s_(a) in QTM
dist pos modM modM+inv
---- -------------- ------------- -------------
0 1 0 0
2 6 0 0
4 27 3 2
6 120 12 10
8 727 72 48
9 32 4 4
10 4,164 479 290
11 1,008 126 65
12 32,163 3,790 2,168
13 16,368 2,046 1,084
14 316,452 38,685 20,965
15 406,704 50,827 25,984
16 4,822,181 599,096 311,322
17 16,168,084 2,020,845 1,020,096
18 139,213,210 17,382,019 8,794,229
19 782,908,280 97,861,909 49,127,729
20 4,065,210,281 508,003,625 255,202,212
21 8,026,900,428 1,003,353,303 503,169,354
22 4,963,326,423 620,211,029 312,119,302
23 346,541,944 43,316,508 21,865,028
24 17,090 1,702 1,548
26 3 0 0
---- -------------- ------------- -------------
tot 18,345,885,696 2,292,846,080 1,151,661,440
Summary of C_s_(b) in QTM
dist pos modM modM+inv
---- ----------- ----------- ----------
0 1 0 0
2 2 0 0
4 1 0 0
6 4 1 1
8 57 11 6
9 16 4 2
10 168 36 27
11 72 18 9
12 1,503 320 204
13 1,056 264 142
14 14,754 3,481 2,042
15 25,170 6,230 3,294
16 209,771 51,682 28,121
17 767,170 191,172 98,222
18 5,101,310 1,272,105 656,218
19 26,570,072 6,637,729 3,373,726
20 110,516,385 27,615,011 13,993,693
21 179,764,414 44,917,282 22,878,230
22 96,491,977 24,109,621 12,278,007
23 5,208,670 1,298,685 703,519
24 706 140 137
26 1 0 0
---- ----------- ----------- ----------
tot 424,673,280 106,103,792 54,015,600
Summary of D_2_(edge) in QTM
dist pos modM modM+inv
---- ------ ------ --------
0 1 0 0
2 2 0 0
4 1 0 0
8 49 10 6
10 130 27 23
12 555 111 86
14 1,234 261 204
16 3,621 816 637
18 12,456 2,893 2,173
20 49,553 11,883 8,381
22 30,615 7,217 5,196
24 86 14 14
26 1 0 0
---- ------ ------ --------
tot 98,304 23,232 16,720
Summary of D_2_(face) in QTM
dist pos modM modM+inv
---- ------- ------ --------
0 1 0 0
2 6 0 0
4 27 2 2
6 120 10 7
8 471 36 27
10 1,686 138 85
12 4,797 372 229
14 11,472 921 563
16 25,313 2,022 1,194
18 77,460 6,222 3,623
20 141,809 11,331 6,589
22 31,707 2,301 1,432
24 40 1 1
26 3 0 0
---- ------- ------ --------
tot 294,912 23,356 13,752
Summary of D_2d_(edge) in QTM
dist pos modM modM+inv
---- ----- ----- --------
0 1 0 0
4 1 0 0
8 9 4 4
10 12 6 6
12 63 25 21
14 92 41 34
16 149 66 63
18 456 220 191
20 1,245 604 456
22 1,023 497 368
24 20 9 9
26 1 0 0
---- ----- ----- --------
tot 3,072 1,472 1,152
Summary of D_2d_(face) in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
4 1 0 0
8 1 0 0
10 2 1 1
12 25 6 6
14 66 28 25
16 55 19 16
18 110 47 43
20 157 60 55
22 87 29 28
24 6 2 2
26 1 0 0
---- --- ---- --------
tot 512 192 176
Summary of D_2h_(edge) in QTM
dist pos modM modM+inv
---- ----- ---- --------
0 1 0 0
4 1 0 0
8 1 0 0
10 2 1 1
12 31 9 9
14 68 29 29
16 179 81 81
18 382 183 183
20 673 318 318
22 699 335 335
24 10 4 4
26 1 0 0
---- ----- ---- --------
tot 2,048 960 960
Summary of D_2h_(face) in QTM
dist pos modM modM+inv
---- ------ ----- --------
0 1 0 0
4 3 0 0
8 39 6 6
12 199 27 27
14 132 17 17
16 797 124 122
18 2,324 379 373
20 5,037 820 684
22 3,743 608 442
24 10 1 1
26 3 0 0
---- ------ ----- --------
tot 12,288 1,982 1,672
Summary of D_3 in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
12 5 2 2
16 6 3 3
17 1 0 0
18 23 11 9
19 31 15 10
20 49 23 20
21 132 63 43
22 125 62 52
23 52 26 19
24 7 3 2
---- --- ---- --------
tot 432 208 160
Summary of D_3d in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
12 1 0 0
17 1 1 1
18 1 1 1
19 1 1 1
20 3 2 2
21 6 6 6
22 1 1 1
24 1 0 0
---- --- ---- --------
tot 16 12 12
Summary of D_4 in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
2 2 1 1
4 1 0 0
8 1 0 0
10 8 4 3
12 43 15 14
14 50 20 17
16 63 23 20
18 78 31 30
20 177 70 63
22 83 27 27
24 4 1 1
26 1 0 0
---- --- ---- --------
tot 512 192 176
Summary of D_4h in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
4 1 1 1
8 1 1 1
12 13 12 12
14 10 10 10
16 17 17 17
18 16 16 16
20 37 36 36
22 29 29 29
24 2 1 1
26 1 1 1
---- --- ---- --------
tot 128 124 124
Summary of O_h in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 1 1
12 1 1 1
20 1 1 1
24 1 1 1
---- --- ---- --------
tot 4 4 4
Summary of S_4 in QTM
dist pos modM modM+inv
---- ------- ------- --------
0 1 0 0
2 2 0 0
4 1 0 0
8 77 17 17
10 72 13 10
11 40 10 5
12 397 80 62
13 128 32 17
14 2,982 692 473
15 788 197 107
16 6,665 1,611 1,022
17 13,396 3,349 1,887
18 42,878 10,543 6,073
19 66,764 16,691 8,944
20 109,157 26,825 15,621
21 132,664 33,166 17,756
22 58,879 14,292 8,795
23 7,404 1,851 1,044
24 72 7 7
26 1 0 0
---- ------- ------- --------
tot 442,368 109,376 61,840
Summary of S_6 in QTM
dist pos modM modM+inv
---- ----- ----- --------
0 1 0 0
8 2 1 1
12 3 1 1
14 14 7 6
15 4 2 1
16 48 23 17
17 15 7 4
18 173 85 62
19 275 137 86
20 1,077 533 394
21 2,464 1,229 715
22 2,521 1,256 944
23 1,130 565 355
24 49 24 24
---- ----- ----- --------
tot 7,776 3,870 2,610
Summary of T in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
12 9 2 2
16 2 0 0
18 6 1 1
20 29 5 5
22 24 4 4
24 1 0 0
---- --- ---- --------
tot 72 12 12
Summary of T_h in QTM
dist pos modM modM+inv
---- --- ---- --------
0 1 0 0
12 1 0 0
16 2 1 1
18 2 1 1
20 9 4 3
22 8 4 3
24 1 0 0
---- --- ---- --------
tot 24 10 8
Summary of hard positions in QTM
symm pos modM modM+inv
----------- ------ ----- --------
O_h 1 1 1
D_4h 6 2 2
D_4 6 1 1
C_4v 6 1 1
D_2h_(face) 6 1 1
D_2d_(face) 12 2 2
D_2_(face) 12 1 1
D_2h_(edge) 24 4 4
D_3 24 3 2
D_2d_(edge) 54 9 9
C_4h 60 10 9
C_2v_(a1) 84 7 7
S_4 84 7 7
C_4 156 13 13
D_2_(edge) 168 14 14
S_6 192 24 24
C_2h_(b) 252 21 21
C_2v_(b) 360 30 29
C_2h_(a) 1,260 105 103
C_3 1,584 99 94
C_s_(b) 3,360 140 137
C_2v_(a2) 4,284 357 309
C_2_(b) 4,776 199 191
C_2_(a) 8,616 359 332
C_i 12,624 526 465
C_s_(a) 40,848 1,702 1,548
----------- ------ ----- --------
tot 78,859 3,638 3,327