metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊4D4, C23.19D6, C3⋊D4⋊C4, C3⋊2(C4×D4), D6⋊2(C2×C4), C2.2(S3×D4), D6⋊C4⋊10C2, C22⋊C4⋊7S3, C22⋊2(C4×S3), C6.18(C2×D4), (C2×C4).28D6, Dic3⋊C4⋊9C2, Dic3⋊1(C2×C4), C6.7(C22×C4), (C4×Dic3)⋊11C2, C6.22(C4○D4), (C2×C6).22C23, C2.2(D4⋊2S3), (C2×C12).51C22, Dic3○2(C22⋊C4), (C22×Dic3)⋊1C2, C22.14(C22×S3), (C22×C6).11C22, (C22×S3).16C22, (C2×Dic3).47C22, (S3×C2×C4)⋊9C2, C2.9(S3×C2×C4), (C2×C6)⋊2(C2×C4), (C3×C22⋊C4)⋊9C2, (C2×C3⋊D4).2C2, C22⋊C4○(C2×Dic3), 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, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, Dic3⋊4D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S3×C2×C4, S3×D4, 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 C4×S3 |
| 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 C4×S3 |
| 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 C4×S3 |
| 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 C4×S3 |
| 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 C4○D4 |
| 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 C4○D4 |
| 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 S3×D4 |
| 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 C4×D4⋊2S3 C42.104D6 C4×S3×D4 C42⋊13D6 C42.108D6 Dic6⋊23D4 C42.119D6 C24.67D6 C24⋊8D6 C24.44D6 C24.46D6 C12⋊(C4○D4) 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(C4×D4) Dic15⋊9D4 C15⋊28(C4×D4) Dic15⋊17D4 Dic15⋊19D4 C3⋊D4⋊F5
Dic3⋊4D4 is a maximal quotient of
C6.(C4×Q8) Dic3⋊C42 C6.(C4×D4) 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(C4×S3) M4(2).22D6 C42.196D6 Dic3×C22⋊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(C4×D4) Dic15⋊9D4 C15⋊28(C4×D4) 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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