metacyclic, supersoluble, monomial, Z-group
Aliases: C37⋊C12, D37.C6, C37⋊C4⋊C3, C37⋊C3⋊C4, C37⋊C6.C2, SmallGroup(444,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊C12 |
Generators and relations for C37⋊C12
G = < a,b | a37=b12=1, bab-1=a23 >
Character table of C37⋊C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 37A | 37B | 37C | |
| size | 1 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | 1 | 1 | 1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | 1 | 1 | 1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | 1 | 1 | 1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | 1 | 1 | 1 | linear of order 12 |
| ρ13 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3734+ζ3733+ζ3732+ζ3730+ζ3724+ζ3719+ζ3718+ζ3713+ζ377+ζ375+ζ374+ζ373 | ζ3736+ζ3731+ζ3729+ζ3727+ζ3726+ζ3723+ζ3714+ζ3711+ζ3710+ζ378+ζ376+ζ37 | ζ3735+ζ3728+ζ3725+ζ3722+ζ3721+ζ3720+ζ3717+ζ3716+ζ3715+ζ3712+ζ379+ζ372 | orthogonal faithful |
| ρ14 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3735+ζ3728+ζ3725+ζ3722+ζ3721+ζ3720+ζ3717+ζ3716+ζ3715+ζ3712+ζ379+ζ372 | ζ3734+ζ3733+ζ3732+ζ3730+ζ3724+ζ3719+ζ3718+ζ3713+ζ377+ζ375+ζ374+ζ373 | ζ3736+ζ3731+ζ3729+ζ3727+ζ3726+ζ3723+ζ3714+ζ3711+ζ3710+ζ378+ζ376+ζ37 | orthogonal faithful |
| ρ15 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3736+ζ3731+ζ3729+ζ3727+ζ3726+ζ3723+ζ3714+ζ3711+ζ3710+ζ378+ζ376+ζ37 | ζ3735+ζ3728+ζ3725+ζ3722+ζ3721+ζ3720+ζ3717+ζ3716+ζ3715+ζ3712+ζ379+ζ372 | ζ3734+ζ3733+ζ3732+ζ3730+ζ3724+ζ3719+ζ3718+ζ3713+ζ377+ζ375+ζ374+ζ373 | orthogonal faithful |
►On 37 points: primitive
Generators in S37
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37)
(2 30 28 7 27 15 37 9 11 32 12 24)(3 22 18 13 16 29 36 17 21 26 23 10)(4 14 8 19 5 6 35 25 31 20 34 33)
G:=sub<Sym(37)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,30,28,7,27,15,37,9,11,32,12,24)(3,22,18,13,16,29,36,17,21,26,23,10)(4,14,8,19,5,6,35,25,31,20,34,33)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,30,28,7,27,15,37,9,11,32,12,24)(3,22,18,13,16,29,36,17,21,26,23,10)(4,14,8,19,5,6,35,25,31,20,34,33) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)], [(2,30,28,7,27,15,37,9,11,32,12,24),(3,22,18,13,16,29,36,17,21,26,23,10),(4,14,8,19,5,6,35,25,31,20,34,33)]])
Matrix representation of C37⋊C12 ►in GL12(𝔽1777)
| 133 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 129 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1731 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 309 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1469 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 45 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1469 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 309 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1731 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 129 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 133 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1776 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 394 | 1613 | 661 | 0 | 0 | 1116 | 164 | 1383 | 0 | 133 | 182 | 130 |
| 383 | 1231 | 902 | 0 | 0 | 921 | 450 | 821 | 0 | 129 | 1511 | 399 |
| 1264 | 1559 | 1653 | 0 | 1 | 1509 | 1399 | 213 | 0 | 1731 | 1600 | 1772 |
| 568 | 923 | 242 | 0 | 0 | 1495 | 1505 | 425 | 0 | 309 | 884 | 1338 |
| 1555 | 1674 | 895 | 0 | 0 | 577 | 356 | 1776 | 0 | 1469 | 1685 | 405 |
| 599 | 1651 | 648 | 0 | 0 | 1567 | 1073 | 1219 | 0 | 45 | 306 | 1339 |
| 1389 | 1742 | 253 | 0 | 0 | 1566 | 1470 | 1349 | 0 | 1469 | 1339 | 306 |
| 1735 | 317 | 800 | 0 | 0 | 1415 | 630 | 83 | 1 | 309 | 405 | 1685 |
| 866 | 1276 | 683 | 0 | 0 | 789 | 754 | 688 | 0 | 1731 | 1338 | 884 |
| 779 | 1069 | 1692 | 0 | 0 | 45 | 1359 | 214 | 0 | 129 | 1772 | 1600 |
| 920 | 1231 | 1536 | 1 | 0 | 1626 | 1727 | 557 | 0 | 133 | 399 | 1511 |
| 1729 | 938 | 81 | 0 | 0 | 1742 | 743 | 1252 | 0 | 1776 | 130 | 182 |
G:=sub<GL(12,GF(1777))| [133,129,1731,309,1469,45,1469,309,1731,129,133,1776,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[394,383,1264,568,1555,599,1389,1735,866,779,920,1729,1613,1231,1559,923,1674,1651,1742,317,1276,1069,1231,938,661,902,1653,242,895,648,253,800,683,1692,1536,81,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1116,921,1509,1495,577,1567,1566,1415,789,45,1626,1742,164,450,1399,1505,356,1073,1470,630,754,1359,1727,743,1383,821,213,425,1776,1219,1349,83,688,214,557,1252,0,0,0,0,0,0,0,1,0,0,0,0,133,129,1731,309,1469,45,1469,309,1731,129,133,1776,182,1511,1600,884,1685,306,1339,405,1338,1772,399,130,130,399,1772,1338,405,1339,306,1685,884,1600,1511,182] >;
C37⋊C12 in GAP, Magma, Sage, TeX
C_{37}\rtimes C_{12} % in TeX
G:=Group("C37:C12"); // GroupNames label
G:=SmallGroup(444,7);
// by ID
G=gap.SmallGroup(444,7);
# by ID
G:=PCGroup([4,-2,-3,-2,-37,24,5955,2503,1163]);
// Polycyclic
G:=Group<a,b|a^37=b^12=1,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C37⋊C12 in TeX
Character table of C37⋊C12 in TeX