metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:3D4, C12:2D4, C23.13D6, (C6xD4):3C2, (C2xD4):4S3, C4:2(C3:D4), C3:4(C4:D4), (C2xC4).51D6, C6.50(C2xD4), C2.26(S3xD4), C4:Dic3:14C2, C6.31(C4oD4), (C2xC6).53C23, C6.D4:11C2, (C2xC12).34C22, C2.17(D4:2S3), (C22xC6).20C22, C22.60(C22xS3), (C22xS3).26C22, (C2xDic3).19C22, (S3xC2xC4):2C2, (C2xC3:D4):5C2, C2.14(C2xC3:D4), SmallGroup(96,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:3D4
G = < a,b,c,d | a6=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 226 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C4:D4, C4:Dic3, C6.D4, S3xC2xC4, C2xC3:D4, C6xD4, D6:3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, S3xD4, D4:2S3, C2xC3:D4, D6:3D4
Character table of D6:3D4
| 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 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 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 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 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 |
| ρ11 | 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 |
| 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 S3 |
| ρ14 | 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 |
| 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 D6 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 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 | -√-3 | -√-3 | √-3 | √-3 | 1 | -1 | complex lifted from C3:D4 |
| ρ18 | 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 |
| 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 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 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 | -2i | 2i | 0 | 0 | 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 | symplectic lifted from D4:2S3, 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 6)(2 5)(3 4)(8 12)(9 11)(13 14)(15 18)(16 17)(19 21)(22 24)(25 29)(26 28)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)(43 45)(46 48)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
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), (1,6)(2,5)(3,4)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)(22,24)(25,29)(26,28)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(43,45)(46,48), (1,41,17,34)(2,42,18,35)(3,37,13,36)(4,38,14,31)(5,39,15,32)(6,40,16,33)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)>;
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), (1,6)(2,5)(3,4)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)(22,24)(25,29)(26,28)(31,36)(32,35)(33,34)(37,38)(39,42)(40,41)(43,45)(46,48), (1,41,17,34)(2,42,18,35)(3,37,13,36)(4,38,14,31)(5,39,15,32)(6,40,16,33)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42) );
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)], [(1,6),(2,5),(3,4),(8,12),(9,11),(13,14),(15,18),(16,17),(19,21),(22,24),(25,29),(26,28),(31,36),(32,35),(33,34),(37,38),(39,42),(40,41),(43,45),(46,48)], [(1,41,17,34),(2,42,18,35),(3,37,13,36),(4,38,14,31),(5,39,15,32),(6,40,16,33),(7,27,44,20),(8,28,45,21),(9,29,46,22),(10,30,47,23),(11,25,48,24),(12,26,43,19)], [(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,15),(8,16),(9,17),(10,18),(11,13),(12,14),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42)]])
D6:3D4 is a maximal subgroup of
D6.D8 D6:D8 D6.SD16 D6:SD16 D6:C8:11C2 C3:C8:1D4 C3:C8:D4 C24:1C4:C2 D12:D4 D6:3D8 Dic6:D4 C24:12D4 D6:8SD16 C24:14D4 D12:7D4 C24:8D4 C42.228D6 D12:24D4 C42.229D6 C42.113D6 C42.115D6 C42.116D6 C42.117D6 C24:7D6 C24.44D6 C24.46D6 C24.47D6 S3xC4:D4 C4:C4:21D6 C6.382+ 1+4 C6.722- 1+4 C6.732- 1+4 D12:20D4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C6.452+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.492+ 1+4 C6.612+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.692+ 1+4 D12:10D4 Dic6:10D4 C42.234D6 C42.144D6 C42.238D6 D12:11D4 Dic6:11D4 C42.168D6 D4xC3:D4 C24.52D6 C24.53D6 (C2xD4):43D6 C6.1072- 1+4 C6.1082- 1+4 C6.1482+ 1+4 C36:2D4 D6:2D12 C12:2D12 C62.100C23 C62.112C23 C62.256C23 C60:4D4 C12:2D20 D30:6D4 (S3xC10):D4 C60:2D4
D6:3D4 is a maximal quotient of
C24.14D6 C24.17D6 C24.23D6 C24.27D6 C12:(C4:C4) (C2xDic3).Q8 C4:(D6:C4) (C2xC12).289D4 C42.61D6 D12.23D4 C12:2D8 Dic6:9D4 C12:5SD16 C12:Q16 D6:3D8 C24:12D4 C24.23D4 C24:14D4 C24:8D4 C24.44D4 D6:3Q16 C24.36D4 C24.29D4 C24.30D6 C24.31D6 C24.32D6 C36:2D4 D6:2D12 C12:2D12 C62.100C23 C62.112C23 C62.256C23 C60:4D4 C12:2D20 D30:6D4 (S3xC10):D4 C60:2D4
Matrix representation of D6:3D4 ►in GL4(F13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 9 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 5 | 0 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,12,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,8],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,0] >;
D6:3D4 in GAP, Magma, Sage, TeX
D_6\rtimes_3D_4
% in TeX
G:=Group("D6:3D4"); // GroupNames label
G:=SmallGroup(96,145);
// by ID
G=gap.SmallGroup(96,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations
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