metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3:4D4, C23.19D6, C3:D4:C4, C3:2(C4xD4), D6:2(C2xC4), C2.2(S3xD4), D6:C4:10C2, C22:C4:7S3, C22:2(C4xS3), C6.18(C2xD4), (C2xC4).28D6, Dic3:C4:9C2, Dic3:1(C2xC4), C6.7(C22xC4), (C4xDic3):11C2, C6.22(C4oD4), (C2xC6).22C23, C2.2(D4:2S3), (C2xC12).51C22, Dic3o2(C22:C4), (C22xDic3):1C2, C22.14(C22xS3), (C22xC6).11C22, (C22xS3).16C22, (C2xDic3).47C22, (S3xC2xC4):9C2, C2.9(S3xC2xC4), (C2xC6):2(C2xC4), (C3xC22:C4):9C2, (C2xC3:D4).2C2, C22:C4o(C2xDic3), SmallGroup(96,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:4D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 202 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C22xS3, C22xC6, C4xD4, C4xDic3, Dic3:C4, D6:C4, C3xC22:C4, S3xC2xC4, C22xDic3, C2xC3:D4, Dic3:4D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, C4xS3, C22xS3, C4xD4, S3xC2xC4, S3xD4, D4:2S3, Dic3:4D4
Character table of Dic3:4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 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 | 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 | -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 | 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 | -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 | 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 | -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 | 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 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -i | -i | i | i | -i | -1 | i | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | i | -i | -i | -i | -i | i | i | i | -1 | -i | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | -i | i | i | -i | -i | i | i | i | 1 | -i | -1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | -i | i | i | -i | 1 | i | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | i | i | -i | -i | i | -1 | -i | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | -i | i | i | i | i | -i | -i | -i | -1 | i | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | i | -i | -i | i | i | -i | -i | -i | 1 | i | -1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | i | -i | -i | i | 1 | -i | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 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 | 2 | -2 | 0 | 0 | 0 | 0 | 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 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | complex lifted from C4xS3 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | complex lifted from C4xS3 |
| ρ25 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | i | complex lifted from C4xS3 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | i | -i | i | -i | complex lifted from C4xS3 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ30 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 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 20 4 23)(2 19 5 22)(3 24 6 21)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 25 16 28)(14 30 17 27)(15 29 18 26)(31 45 34 48)(32 44 35 47)(33 43 36 46)
(1 37 16 35)(2 42 17 34)(3 41 18 33)(4 40 13 32)(5 39 14 31)(6 38 15 36)(7 26 43 24)(8 25 44 23)(9 30 45 22)(10 29 46 21)(11 28 47 20)(12 27 48 19)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,38,10,41)(8,37,11,40)(9,42,12,39)(13,25,16,28)(14,30,17,27)(15,29,18,26)(31,45,34,48)(32,44,35,47)(33,43,36,46), (1,37,16,35)(2,42,17,34)(3,41,18,33)(4,40,13,32)(5,39,14,31)(6,38,15,36)(7,26,43,24)(8,25,44,23)(9,30,45,22)(10,29,46,21)(11,28,47,20)(12,27,48,19), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)>;
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,38,10,41)(8,37,11,40)(9,42,12,39)(13,25,16,28)(14,30,17,27)(15,29,18,26)(31,45,34,48)(32,44,35,47)(33,43,36,46), (1,37,16,35)(2,42,17,34)(3,41,18,33)(4,40,13,32)(5,39,14,31)(6,38,15,36)(7,26,43,24)(8,25,44,23)(9,30,45,22)(10,29,46,21)(11,28,47,20)(12,27,48,19), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48) );
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)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,38,10,41),(8,37,11,40),(9,42,12,39),(13,25,16,28),(14,30,17,27),(15,29,18,26),(31,45,34,48),(32,44,35,47),(33,43,36,46)], [(1,37,16,35),(2,42,17,34),(3,41,18,33),(4,40,13,32),(5,39,14,31),(6,38,15,36),(7,26,43,24),(8,25,44,23),(9,30,45,22),(10,29,46,21),(11,28,47,20),(12,27,48,19)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,31),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)]])
Dic3:4D4 is a maximal subgroup of
C24.35D6 C24.38D6 C24.42D6 C42.188D6 C42.91D6 C42:12D6 C42.96D6 C4xD4:2S3 C42.104D6 C4xS3xD4 C42:13D6 C42.108D6 Dic6:23D4 C42.119D6 C24.67D6 C24:8D6 C24.44D6 C24.46D6 C12:(C4oD4) Dic6:20D4 C6.342+ 1+4 C4:C4:21D6 C6.402+ 1+4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C4:C4.187D6 C4:C4:26D6 Dic6:21D4 C6.1182+ 1+4 C6.522+ 1+4 C6.562+ 1+4 C6.782- 1+4 C4:C4.197D6 C6.1212+ 1+4 C6.822- 1+4 C4:C4:28D6 C6.1222+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C6.852- 1+4 C42.233D6 C42.137D6 C42.138D6 Dic6:10D4 C42:22D6 C42:23D6 C42.234D6 C42.143D6 C42.160D6 C42.189D6 C42.161D6 C42.162D6 C42.163D6 C42.164D6 Dic9:4D4 C62.49C23 Dic3:4D12 C62.51C23 C62.72C23 C62.94C23 C62.115C23 C62.225C23 Dic3:2S4 Dic3:4D20 Dic15:13D4 C15:17(C4xD4) Dic15:9D4 C15:28(C4xD4) Dic15:17D4 Dic15:19D4 C3:D4:F5
Dic3:4D4 is a maximal quotient of
C6.(C4xQ8) Dic3:C42 C6.(C4xD4) Dic3:C4:C4 D6:C42 D6:C4:C4 D6:C4:5C4 D6:C4:3C4 C3:D4:C8 D6:2M4(2) Dic3:M4(2) C3:C8:26D4 Dic3:4D8 D4.S3:C4 Dic3:6SD16 D4:S3:C4 Dic3:7SD16 C3:Q16:C4 Dic3:4Q16 Q8:3(C4xS3) M4(2).22D6 C42.196D6 Dic3xC22:C4 C24.14D6 C24.15D6 C24.57D6 C24.23D6 C24.24D6 C24.60D6 Dic9:4D4 C62.49C23 Dic3:4D12 C62.51C23 C62.72C23 C62.94C23 C62.115C23 C62.225C23 Dic3:4D20 Dic15:13D4 C15:17(C4xD4) Dic15:9D4 C15:28(C4xD4) Dic15:17D4 Dic15:19D4 C3:D4:F5
Matrix representation of Dic3:4D4 ►in GL4(F13) generated by
| 1 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,1,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[0,8,0,0,8,0,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,11,1] >;
Dic3:4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_4D_4 % in TeX
G:=Group("Dic3:4D4"); // GroupNames label
G:=SmallGroup(96,88);
// by ID
G=gap.SmallGroup(96,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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