| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 3.12.20.1 |
$12$ |
$x^{12} - 144 x^{10} + 12 x^{9} + 7776 x^{8} - 139266 x^{6} + 797040 x^{4} + 423900 x^{3} + 1609632 x^{2} + 287712 x + 110889$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^2:C_4\times S_3$ (as 12T119) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + 18 t^{3} x + 18 t^{3} + 18 t^{2} + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.10 |
$12$ |
$x^{12} - 72 x^{11} + 2430 x^{10} - 29112 x^{9} + 235305 x^{8} + 589356 x^{7} + 808758 x^{6} + 829116 x^{5} + 362070 x^{4} - 59184 x^{3} - 45684 x^{2} + 72900 x + 50625$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^2:C_4\times S_3$ (as 12T119) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 9 t^{2} + 9\right) x^{2} + \left(9 t^{2} + 18 t\right) x + 9 t^{3} + 9 t^{2} + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.11 |
$12$ |
$x^{12} - 18 x^{11} + 288 x^{10} - 3642 x^{9} + 66825 x^{8} + 489564 x^{7} + 1863378 x^{6} + 4339494 x^{5} + 6999615 x^{4} + 7773678 x^{3} + 5967108 x^{2} + 2842128 x + 737586$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 9 t^{2} + 9 t + 9\right) x^{2} + \left(9 t^{3} + 9 t^{2} + 18 t + 9\right) x + 18 t^{3} + 9 t^{2} + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.12 |
$12$ |
$x^{12} - 180 x^{11} + 10404 x^{10} - 121398 x^{9} + 2903202 x^{8} - 678618 x^{7} + 1727460 x^{6} + 3247938 x^{5} + 4136589 x^{4} - 3041280 x^{3} - 3189132 x^{2} + 696438 x + 578421$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 18 t\right) x^{2} + 9 x + 9 t^{2} + 9 t + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.13 |
$12$ |
$x^{12} + 72 x^{11} + 2160 x^{10} + 23178 x^{9} - 36774 x^{8} - 1448118 x^{7} + 1360692 x^{6} + 12416814 x^{5} + 18786573 x^{4} + 15404364 x^{3} + 9738468 x^{2} + 4293810 x + 842805$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{2} + 18 t + 18\right) x^{2} + \left(18 t^{3} + 9 t^{2} + 18 t + 9\right) x + 18 t^{3} + 9 t + 3$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.14 |
$12$ |
$x^{12} - 36 x^{11} + 648 x^{10} - 5100 x^{9} + 12960 x^{8} + 120528 x^{7} + 117180 x^{6} - 46656 x^{5} + 104976 x^{4} + 263520 x^{3} + 188568 x^{2} + 244944 x + 155844$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 9 t^{2}\right) x^{2} + \left(18 t^{3} + 18 t^{2} + 18\right) x + 9 t^{2} + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.15 |
$12$ |
$x^{12} + 54 x^{10} + 12 x^{9} + 972 x^{8} + 5400 x^{6} - 11664 x^{5} - 10206 x^{4} - 43632 x^{3} + 34992 x^{2} + 233280 x + 168480$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t + 18\right) x + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.16 |
$12$ |
$x^{12} - 72 x^{10} - 10986 x^{9} + 409374 x^{8} + 2312550 x^{7} + 5136912 x^{6} + 6875442 x^{5} + 5981445 x^{4} + 2410182 x^{3} + 381996 x^{2} - 121500 x + 61074$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^2:C_4\times S_3$ (as 12T119) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 9 t^{2} + 18 t + 18\right) x^{2} + \left(18 t^{3} + 18 t^{2} + 18 t + 9\right) x + 9 t^{3} + 18 t^{2} + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.17 |
$12$ |
$x^{12} - 108 x^{10} - 12012 x^{9} + 420714 x^{8} + 3115908 x^{7} + 8387604 x^{6} + 11429748 x^{5} + 9035469 x^{4} + 4043628 x^{3} + 1043928 x^{2} + 169128 x + 18792$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 9 t^{2} + 18 t + 18\right) x^{2} + \left(18 t^{3} + 9 t^{2} + 18 t + 9\right) x + 9 t^{3} + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.18 |
$12$ |
$x^{12} - 90 x^{11} + 3798 x^{10} - 81024 x^{9} + 925830 x^{8} + 3175200 x^{7} + 5644296 x^{6} + 6774840 x^{5} + 6191154 x^{4} + 3846312 x^{3} + 1990656 x^{2} + 544320 x + 129600$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 18 t^{2} + 9 t\right) x^{2} + \left(9 t^{3} + 9 t\right) x + 18 t^{3} + 18 t^{2} + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.19 |
$12$ |
$x^{12} + 36 x^{11} + 882 x^{10} + 5574 x^{9} + 57429 x^{8} + 202500 x^{7} + 912600 x^{6} + 1587276 x^{5} + 2101707 x^{4} + 1960146 x^{3} + 1260684 x^{2} + 602640 x + 146610$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{2} + 18 t + 9\right) x^{2} + \left(9 t^{2} + 9\right) x + 9 t^{2} + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.2 |
$12$ |
$x^{12} - 18 x^{11} + 216 x^{10} - 2166 x^{9} + 53379 x^{8} + 642492 x^{7} + 2731212 x^{6} + 5988006 x^{5} + 7345404 x^{4} + 4946616 x^{3} + 1664550 x^{2} + 224532 x + 15633$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_6.D_6$ (as 12T39) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 9 t^{2} + 9 t + 9\right) x^{2} + \left(18 t^{3} + 18 t^{2} + 18 t\right) x + 9 t^{3} + 3$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.20 |
$12$ |
$x^{12} - 36 x^{11} + 72 x^{10} - 1896 x^{9} + 210600 x^{8} - 32400 x^{7} - 22464 x^{6} - 178848 x^{5} - 283824 x^{4} + 6912 x^{3} + 217728 x^{2} + 124416 x + 20736$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + 18 t x^{2} + 18 x + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.21 |
$12$ |
$x^{12} - 90 x^{11} + 3258 x^{10} - 37176 x^{9} + 836649 x^{8} + 1762884 x^{7} + 909036 x^{6} + 878688 x^{5} + 1212084 x^{4} - 320166 x^{3} - 690606 x^{2} + 166698 x + 194481$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^2:C_4\times S_3$ (as 12T119) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 9 t^{2} + 9 t + 9\right) x^{2} + \left(9 t^{3} + 9 t^{2} + 9 t + 18\right) x + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.22 |
$12$ |
$x^{12} - 72 x^{11} + 2412 x^{10} - 28266 x^{9} + 231903 x^{8} + 324648 x^{7} + 249534 x^{6} + 387342 x^{5} + 432054 x^{4} - 299754 x^{3} - 526824 x^{2} + 14094 x + 369441$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_6.D_6$ (as 12T39) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 9 t^{2} + 9\right) x^{2} + 9 t x + 9 t^{3} + 18 t^{2} + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.23 |
$12$ |
$x^{12} - 90 x^{11} + 3096 x^{10} - 26718 x^{9} + 681291 x^{8} + 2423196 x^{7} + 5699376 x^{6} + 7319646 x^{5} + 6384096 x^{4} + 6514452 x^{3} + 5675022 x^{2} + 2439720 x + 411237$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 9 t^{2} + 9 t + 9\right) x^{2} + 18 t^{3} x + 9 t^{3} + 9 t^{2} + 18 t + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.24 |
$12$ |
$x^{12} - 72 x^{11} + 2682 x^{10} - 66102 x^{9} + 1732995 x^{8} + 2934306 x^{7} + 4585626 x^{6} + 4155300 x^{5} + 3313548 x^{4} + 1843020 x^{3} + 954504 x^{2} + 295488 x + 74682$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_6.D_6$ (as 12T39) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 18 t^{2} + 18 t + 9\right) x^{2} + 9 t^{3} x + 9 t^{3} + 9 t^{2} + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.25 |
$12$ |
$x^{12} - 6 x^{11} + 18 x^{10} + 6 x^{9} - 15 x^{6} + 252 x^{5} + 162 x^{4} - 72 x^{3} + 162 x^{2} + 585$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 t x^{5} + 3 x^{3} + \left(9 t + 18\right) x^{2} + 21 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.26 |
$12$ |
$x^{12} - 12 x^{11} + 72 x^{10} - 18 x^{8} + 234 x^{7} - 114 x^{6} + 180 x^{5} + 162 x^{4} - 162 x^{3} + 351 x^{2} - 54 x + 153$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 t x^{5} + 9 t x^{2} + 9 x + 12 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.27 |
$12$ |
$x^{12} + 6 x^{11} + 9 x^{10} - 36 x^{7} - 105 x^{6} + 90 x^{5} + 432 x^{4} + 774 x^{3} + 1404 x^{2} + 864 x + 450$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{5} + \left(3 t + 3\right) x^{3} + \left(18 t + 18\right) x^{2} + 18 t x + 21 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.28 |
$12$ |
$x^{12} + 6 x^{11} + 18 x^{10} + 6 x^{9} + 54 x^{7} + 21 x^{6} + 405 x^{4} + 36 x^{3} + 45$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(3 t + 6\right) x^{5} + 3 x^{3} + \left(18 t + 9\right) x^{2} + 3 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.29 |
$12$ |
$x^{12} + 36 x^{10} + 12 x^{9} - 36 x^{8} + 216 x^{7} + 264 x^{6} - 180 x^{5} + 648 x^{4} + 720 x^{3} + 216 x^{2} + 108 x + 45$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 6\right) x^{5} + 6 x^{3} + 18 t x^{2} + \left(18 t + 18\right) x + 3 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.3 |
$12$ |
$x^{12} + 522 x^{10} + 1488 x^{9} + 71280 x^{8} + 446634 x^{7} + 2668032 x^{6} + 6828300 x^{5} + 12239667 x^{4} + 15607998 x^{3} + 13676040 x^{2} + 8496738 x + 2906361$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 18 t^{2} + 18\right) x^{2} + \left(18 t^{3} + 9 t + 9\right) x + 18 t^{3} + 9 t^{2} + 18 t + 3$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.30 |
$12$ |
$x^{12} + 6 x^{11} + 9 x^{10} - 6 x^{9} - 18 x^{8} + 24 x^{6} + 72 x^{5} + 81 x^{4} + 90 x^{3} + 324 x^{2} + 333$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\SOPlus(4,2)$ (as 12T34) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{5} + 3 t x^{3} + \left(9 t + 9\right) x^{2} + 18 t + 21$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.31 |
$12$ |
$x^{12} - 6 x^{11} + 18 x^{10} + 6 x^{9} + 36 x^{8} - 108 x^{7} + 96 x^{6} + 90 x^{5} + 378 x^{4} + 126 x^{3} + 513 x^{2} + 162 x + 225$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\PSU(3,2)$ (as 12T47) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 t x^{5} + \left(3 t + 6\right) x^{3} + 18 x^{2} + \left(9 t + 9\right) x + 9 t + 21$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.32 |
$12$ |
$x^{12} + 9 x^{10} + 12 x^{9} + 72 x^{7} + 96 x^{6} + 189 x^{4} + 198 x^{3} + 162 x^{2} + 54 x + 9$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\SOPlus(4,2)$ (as 12T34) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(3 t + 3\right) x^{5} + 6 x^{3} + \left(9 t + 9\right) x^{2} + \left(9 t + 18\right) x + 3$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.33 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} + 54 x^{8} + 252 x^{7} + 186 x^{6} + 360 x^{5} + 513 x^{4} + 594 x^{3} + 594 x^{2} + 270 x + 117$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:Q_8$ (as 12T174) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + \left(6 t + 6\right) x^{3} + \left(18 t + 9\right) x^{2} + 9 t x + 9 t + 3$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.34 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} - 18 x^{8} + 162 x^{7} + 150 x^{6} - 126 x^{5} + 162 x^{4} + 162 x^{3} - 297 x^{2} + 441$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:Q_8$ (as 12T174) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + 9 t x^{2} + \left(9 t + 9\right) x + 21$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.35 |
$12$ |
$x^{12} + 6 x^{11} + 9 x^{10} + 12 x^{9} + 36 x^{8} + 66 x^{6} + 90 x^{5} + 81 x^{4} + 342 x^{3} + 135 x^{2} + 54 x + 234$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{5} + 6 x^{3} + \left(9 t + 9\right) x^{2} + \left(9 t + 9\right) x + 3 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.36 |
$12$ |
$x^{12} - 12 x^{11} + 72 x^{10} + 414 x^{7} + 219 x^{6} + 396 x^{5} + 702 x^{4} + 774 x^{3} + 1026 x^{2} + 432 x + 450$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 t x^{5} + \left(3 t + 3\right) x^{3} + 18 t x^{2} + 9 t x + 21 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.37 |
$12$ |
$x^{12} - 12 x^{11} + 72 x^{10} - 6 x^{9} + 108 x^{8} + 216 x^{7} + 237 x^{6} + 612 x^{5} + 540 x^{4} + 972 x^{3} + 1080 x^{2} + 756 x + 585$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 t x^{5} + \left(6 t + 3\right) x^{3} + \left(18 t + 18\right) x^{2} + \left(18 t + 18\right) x + 21 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.38 |
$12$ |
$x^{12} + 36 x^{10} + 18 x^{8} + 18 x^{7} + 132 x^{6} + 108 x^{5} + 81 x^{4} + 162 x^{3} + 378 x^{2} + 378 x + 225$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\SOPlus(4,2)$ (as 12T34) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 6\right) x^{5} + 9 x^{2} + \left(9 t + 18\right) x + 9 t + 21$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.39 |
$12$ |
$x^{12} + 6 x^{11} + 9 x^{10} - 6 x^{9} + 54 x^{7} + 6 x^{6} - 90 x^{5} + 81 x^{4} + 144 x^{3} - 108 x^{2} + 360$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\SOPlus(4,2)$ (as 12T34) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{5} + 3 t x^{3} + 9 x^{2} + 18 t + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.4 |
$12$ |
$x^{12} + 72 x^{11} + 1998 x^{10} + 32736 x^{9} + 358668 x^{8} + 1188270 x^{7} + 2200338 x^{6} + 2552472 x^{5} + 2111184 x^{4} + 1374516 x^{3} + 809676 x^{2} + 377622 x + 110889$ |
$3$ |
$3$ |
$4$ |
$20$ |
$C_3^3:(C_4\times S_3)$ (as 12T170) |
$4$ |
$2$ |
$[5/2]$ |
$[3/2, 3/2, 3/2, 5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(9 t^{3} + 18 t^{2} + 18\right) x^{2} + \left(9 t^{3} + 18 t^{2} + 9 t + 18\right) x + 18 t^{3} + 18 t^{2} + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.40 |
$12$ |
$x^{12} + 36 x^{10} + 6 x^{9} + 18 x^{8} + 144 x^{7} + 48 x^{6} + 36 x^{5} + 270 x^{4} - 216 x^{3} + 108 x^{2} + 540 x + 234$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 6\right) x^{5} + \left(3 t + 6\right) x^{3} + 9 t x^{2} + 18 x + 3 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.41 |
$12$ |
$x^{12} + 9 x^{10} + 12 x^{9} + 108 x^{7} + 84 x^{6} + 126 x^{5} + 324 x^{4} + 288 x^{3} + 837 x^{2} + 378 x + 450$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(3 t + 3\right) x^{5} + 6 x^{3} + \left(18 t + 18\right) x^{2} + \left(9 t + 9\right) x + 21 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.42 |
$12$ |
$x^{12} + 6 x^{11} + 18 x^{10} + 6 x^{9} + 18 x^{8} + 78 x^{6} + 72 x^{5} + 216 x^{4} + 234 x^{3} + 189 x^{2} + 162 x + 90$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\PSU(3,2)$ (as 12T47) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(3 t + 6\right) x^{5} + \left(3 t + 6\right) x^{3} + 9 t x^{2} + \left(9 t + 9\right) x + 9 t + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.43 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} + 6 x^{9} - 18 x^{8} + 90 x^{7} + 201 x^{6} - 54 x^{5} + 27 x^{4} + 252 x^{3} + 216 x^{2} - 216 x + 234$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + 3 x^{3} + \left(9 t + 9\right) x^{2} + 9 t x + 3 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.44 |
$12$ |
$x^{12} + 36 x^{10} - 6 x^{9} + 36 x^{8} + 198 x^{7} + 210 x^{6} + 252 x^{5} + 486 x^{4} + 792 x^{3} + 837 x^{2} + 648 x + 288$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 6\right) x^{5} + 3 t x^{3} + \left(18 t + 18\right) x^{2} + \left(18 t + 9\right) x + 12 t$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.45 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} - 12 x^{9} + 108 x^{8} + 216 x^{7} + 276 x^{6} + 468 x^{5} + 540 x^{4} + 936 x^{3} + 972 x^{2} + 648 x + 360$ |
$3$ |
$6$ |
$2$ |
$20$ |
$\PSU(3,2)$ (as 12T47) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + 6 t x^{3} + \left(18 t + 18\right) x^{2} + \left(18 t + 18\right) x + 18 t + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.46 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} - 12 x^{9} + 108 x^{8} + 198 x^{7} + 336 x^{6} + 486 x^{5} + 648 x^{4} + 936 x^{3} + 1161 x^{2} + 810 x + 450$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + 6 t x^{3} + \left(18 t + 18\right) x^{2} + \left(18 t + 9\right) x + 21 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.47 |
$12$ |
$x^{12} + 6 x^{11} + 9 x^{10} - 6 x^{9} + 18 x^{8} + 108 x^{7} + 42 x^{6} - 36 x^{5} + 324 x^{4} - 72 x^{3} + 432 x^{2} + 144$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:D_4$ (as 12T172) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{5} + 3 t x^{3} + 18 x^{2} + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.48 |
$12$ |
$x^{12} + 9 x^{10} + 18 x^{8} + 54 x^{7} + 123 x^{6} + 108 x^{5} + 189 x^{4} + 378 x^{3} + 486 x^{2} + 324 x + 90$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:D_4$ (as 12T172) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(3 t + 3\right) x^{5} + \left(3 t + 3\right) x^{3} + \left(9 t + 9\right) x^{2} + \left(18 t + 18\right) x + 9 t + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.49 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} + 18 x^{8} + 36 x^{7} + 264 x^{6} + 162 x^{5} + 162 x^{4} + 162 x^{3} + 567 x^{2} + 486 x + 153$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + \left(9 t + 18\right) x^{2} + \left(18 t + 9\right) x + 12 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.5 |
$12$ |
$x^{12} - 144 x^{11} + 7668 x^{10} - 129660 x^{9} + 708264 x^{8} + 912708 x^{7} + 1567890 x^{6} + 7402752 x^{5} + 19458468 x^{4} + 27085860 x^{3} + 22233528 x^{2} + 10477188 x + 2365929$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + 18 t^{3} x^{2} + \left(18 t^{3} + 18 t + 18\right) x + 18 t^{3} + 9 t^{2} + 18 t + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.50 |
$12$ |
$x^{12} - 6 x^{11} + 18 x^{10} - 6 x^{9} + 90 x^{8} - 72 x^{7} - 33 x^{6} - 180 x^{5} + 216 x^{4} + 594 x^{3} + 864 x^{2} + 540 x + 234$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 t x^{5} + \left(6 t + 3\right) x^{3} + 18 x^{2} + 18 x + 3 t + 18$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.51 |
$12$ |
$x^{12} - 6 x^{11} + 45 x^{10} + 18 x^{8} + 180 x^{7} + 69 x^{6} + 36 x^{5} + 216 x^{4} + 108 x^{2} + 54 x + 9$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:Q_8$ (as 12T174) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 3\right) x^{5} + \left(3 t + 3\right) x^{3} + 9 t x^{2} + \left(9 t + 18\right) x + 3$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.52 |
$12$ |
$x^{12} + 12 x^{11} + 36 x^{10} - 18 x^{8} - 108 x^{7} + 15 x^{6} + 144 x^{5} + 459 x^{4} + 378 x^{3} + 351 x^{2} + 162 x + 90$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:D_4$ (as 12T172) |
$2$ |
$4$ |
$[9/4]$ |
$[5/4, 5/4, 9/4, 9/4]_{4}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 x^{5} + \left(3 t + 3\right) x^{3} + \left(18 t + 9\right) x^{2} + \left(9 t + 9\right) x + 9 t + 12$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.53 |
$12$ |
$x^{12} + 12 x^{11} + 36 x^{10} + 12 x^{6} + 72 x^{5} + 324 x^{4} + 108 x^{2} + 45$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 x^{5} + \left(18 t + 18\right) x^{2} + 3 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.12.20.54 |
$12$ |
$x^{12} - 12 x^{11} + 72 x^{10} + 72 x^{8} + 126 x^{7} - 60 x^{6} + 72 x^{5} + 81 x^{4} + 36 x^{3} + 135 x^{2} + 108 x + 45$ |
$3$ |
$6$ |
$2$ |
$20$ |
$C_3^4:\OD_{16}$ (as 12T215) |
$4$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4, 9/4, 9/4]_{4}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 6 t x^{5} + \left(6 t + 6\right) x^{3} + \left(9 t + 9\right) x^{2} + 9 x + 3 t + 9$ |
$[5, 0]$ |
$[1, 1]$ |
