metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:2D4, C4:2D12, C12:1D4, C4:C4:3S3, D6:C4:8C2, C6.7(C2xD4), (C2xD12):4C2, C3:2(C4:D4), (C2xC4).12D6, C2.13(S3xD4), C2.9(C2xD12), C6.34(C4oD4), (C2xC6).36C23, (C2xC12).5C22, C2.6(Q8:3S3), (C22xS3).7C22, C22.50(C22xS3), (C2xDic3).31C22, (S3xC2xC4):1C2, (C3xC4:C4):6C2, SmallGroup(96,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:D4
G = < a,b,c | a12=b4=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >
Subgroups: 266 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C4xS3, D12, C2xDic3, C2xC12, C2xC12, C22xS3, C22xS3, C4:D4, D6:C4, C3xC4:C4, S3xC2xC4, C2xD12, C2xD12, C12:D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C22xS3, C4:D4, C2xD12, S3xD4, Q8:3S3, C12:D4
Character table of C12:D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 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 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√3 | 1 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √3 | 1 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| 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 Q8:3S3, Schur index 2 |
►On 48 points
(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 22 33)(2 42 23 28)(3 37 24 35)(4 44 13 30)(5 39 14 25)(6 46 15 32)(7 41 16 27)(8 48 17 34)(9 43 18 29)(10 38 19 36)(11 45 20 31)(12 40 21 26)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 24)(12 23)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 48)
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), (1,47,22,33)(2,42,23,28)(3,37,24,35)(4,44,13,30)(5,39,14,25)(6,46,15,32)(7,41,16,27)(8,48,17,34)(9,43,18,29)(10,38,19,36)(11,45,20,31)(12,40,21,26), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,24)(12,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48)>;
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), (1,47,22,33)(2,42,23,28)(3,37,24,35)(4,44,13,30)(5,39,14,25)(6,46,15,32)(7,41,16,27)(8,48,17,34)(9,43,18,29)(10,38,19,36)(11,45,20,31)(12,40,21,26), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,24)(12,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48) );
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)], [(1,47,22,33),(2,42,23,28),(3,37,24,35),(4,44,13,30),(5,39,14,25),(6,46,15,32),(7,41,16,27),(8,48,17,34),(9,43,18,29),(10,38,19,36),(11,45,20,31),(12,40,21,26)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,24),(12,23),(25,29),(26,28),(30,36),(31,35),(32,34),(37,45),(38,44),(39,43),(40,42),(46,48)]])
C12:D4 is a maximal subgroup of
D4:D12 D6:D8 D4:3D12 C3:C8:D4 Q8:3D12 D6:2SD16 Q8:4D12 C3:(C8:D4) D6.4SD16 C8:8D12 C24:7D4 C4.Q8:S3 D6.5D8 D6:2D8 C2.D8:S3 C8:3D12 C6.2- 1+4 C6.2+ 1+4 C6.112+ 1+4 C42:10D6 C42:11D6 C42.95D6 C42.97D6 C42.228D6 D4xD12 D4:5D12 C42.116D6 Q8:6D12 Q8:7D12 C42.131D6 C42.133D6 Dic6:20D4 S3xC4:D4 C6.382+ 1+4 D12:19D4 C4:C4:26D6 C6.172- 1+4 D12:21D4 Dic6:22D4 C6.562+ 1+4 C6.592+ 1+4 C6.1202+ 1+4 C6.1212+ 1+4 C6.662+ 1+4 C6.682+ 1+4 C42.237D6 C42.150D6 C42.153D6 C42.155D6 C42.158D6 C42:25D6 C42.163D6 C42:27D6 C42.240D6 D12:12D4 C42.178D6 C42.179D6 C4:D36 Dic3:D12 C12:7D12 Dic3:3D12 C12:2D12 C12:3D12 Dic5:D12 D30:D4 C60:6D4 C20:2D12 C4:D60
C12:D4 is a maximal quotient of
(C22xC4).85D6 (C2xC4):9D12 D6:C4:3C4 (C2xC12):5D4 C6.C22wrC2 (C2xC4).21D12 C12:SD16 C4:D24 D12.19D4 C42.36D6 Dic6:8D4 C4:Dic12 C8:8D12 C24:7D4 C8.2D12 D6:2D8 C8:3D12 D6:2Q16 C24.18D4 C24.19D4 C24.42D4 C4:C4:6Dic3 C4:(D6:C4) (C2xD12):10C4 (C2xC4):3D12 (C2xC12).56D4 C4:D36 Dic3:D12 C12:7D12 Dic3:3D12 C12:2D12 C12:3D12 Dic5:D12 D30:D4 C60:6D4 C20:2D12 C4:D60
Matrix representation of C12:D4 ►in GL4(F13) generated by
| 1 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 1 | 12 |
| 3 | 7 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
G:=sub<GL(4,GF(13))| [1,1,0,0,12,0,0,0,0,0,1,1,0,0,11,12],[3,6,0,0,7,10,0,0,0,0,12,12,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,12] >;
C12:D4 in GAP, Magma, Sage, TeX
C_{12}\rtimes D_4 % in TeX
G:=Group("C12:D4"); // GroupNames label
G:=SmallGroup(96,102);
// by ID
G=gap.SmallGroup(96,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export