metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12:3D4, Dic3:1D4, C23.15D6, (C2xD4):6S3, (C6xD4):4C2, (C2xD12):9C2, C3:2(C4:1D4), C4:1(C3:D4), (C2xC4).52D6, C6.52(C2xD4), C2.28(S3xD4), (C4xDic3):6C2, (C2xC6).55C23, (C2xC12).35C22, C22.62(C22xS3), (C22xC6).22C22, (C22xS3).12C22, (C2xDic3).39C22, (C2xC3:D4):7C2, C2.16(C2xC3:D4), SmallGroup(96,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:3D4
G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 290 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C2xD4, C2xD4, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C4:1D4, C4xDic3, C2xD12, C2xC3:D4, C6xD4, C12:3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4:1D4, S3xD4, C2xC3:D4, C12:3D4
Character table of C12:3D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
ρ15 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | -√-3 | √-3 | √-3 | 1 | -1 | complex lifted from C3:D4 |
ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | √-3 | -√-3 | -√-3 | 1 | -1 | complex lifted from C3:D4 |
ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | -1 | 1 | complex lifted from C3:D4 |
ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | -1 | 1 | complex lifted from C3:D4 |
ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
(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 38 39 40 41 42 43 44 45 46 47 48)
(1 47 34 24)(2 40 35 17)(3 45 36 22)(4 38 25 15)(5 43 26 20)(6 48 27 13)(7 41 28 18)(8 46 29 23)(9 39 30 16)(10 44 31 21)(11 37 32 14)(12 42 33 19)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 31)(26 30)(27 29)(32 36)(33 35)
G:=sub<Sym(48)| (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35)>;
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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35) );
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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,47,34,24),(2,40,35,17),(3,45,36,22),(4,38,25,15),(5,43,26,20),(6,48,27,13),(7,41,28,18),(8,46,29,23),(9,39,30,16),(10,44,31,21),(11,37,32,14),(12,42,33,19)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47),(25,31),(26,30),(27,29),(32,36),(33,35)]])
C12:3D4 is a maximal subgroup of
C23.2D12 D12:1D4 Dic3.SD16 Dic6:2D4 C4:C4.D6 D12:3D4 Dic3:D8 C24:5D4 C24:11D4 Dic3:5SD16 C24:15D4 C24:9D4 D12:18D4 2+ 1+4:6S3 C42.228D6 Dic6:24D4 C42.114D6 C42.116D6 C24:8D6 C24.45D6 C24.47D6 C12:(C4oD4) Dic6:20D4 C6.382+ 1+4 D12:19D4 C6.442+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C6.682+ 1+4 C42.233D6 C42:20D6 C42.143D6 S3xC4:1D4 C42:28D6 Dic6:11D4 D4xC3:D4 C24.52D6 C6.1462+ 1+4 (C2xC12):17D4 C6.1482+ 1+4 C36:D4 C12:D12 C62.84C23 C62.121C23 C62.258C23 C12:D20 C60:10D4 Dic15:5D4 C60:3D4
C12:3D4 is a maximal quotient of
C24.18D6 C24.24D6 C23:3D12 (C4xDic3):8C4 (C2xD12):10C4 (C2xC12).290D4 C42.64D6 C42.214D6 C42.65D6 C12:D8 C42.74D6 C12:4SD16 C12:6SD16 C42.80D6 C12:3Q16 C24:5D4 C24:11D4 C24.22D4 C24.31D4 C24.43D4 C24:15D4 C24:9D4 C24.26D4 C24.37D4 C24.28D4 C24.30D6 C24.32D6 C36:D4 C12:D12 C62.84C23 C62.121C23 C62.258C23 C12:D20 C60:10D4 Dic15:5D4 C60:3D4
Matrix representation of C12:3D4 ►in GL4(F13) generated by
0 | 12 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | 12 | 1 |
0 | 0 | 12 | 0 |
0 | 12 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | 4 | 11 |
0 | 0 | 2 | 9 |
1 | 0 | 0 | 0 |
0 | 12 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,0,0,0,0,0,4,2,0,0,11,9],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C12:3D4 in GAP, Magma, Sage, TeX
C_{12}\rtimes_3D_4
% in TeX
G:=Group("C12:3D4");
// GroupNames label
G:=SmallGroup(96,147);
// by ID
G=gap.SmallGroup(96,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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