metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊5C4, Dic3⋊5D4, C3⋊3(C4×D4), C4⋊C4⋊8S3, C4⋊1(C4×S3), C12⋊2(C2×C4), D6⋊3(C2×C4), C2.4(S3×D4), D6⋊C4⋊12C2, C6.24(C2×D4), (C2×C4).31D6, Dic3○3(C4⋊C4), (C4×Dic3)⋊3C2, (C2×D12).7C2, C6.33(C4○D4), (C2×C6).34C23, C6.11(C22×C4), (C2×C12).24C22, C2.2(Q8⋊3S3), C22.18(C22×S3), (C22×S3).20C22, (C2×Dic3).49C22, (C3×C4⋊C4)⋊4C2, (S3×C2×C4)⋊12C2, C2.13(S3×C2×C4), C4⋊C4○(C2×Dic3), SmallGroup(96,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 218 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×D4, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, Dic3⋊5D4
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, Q8⋊3S3, Dic3⋊5D4
Character table of Dic3⋊5D4
| 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 | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 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 | 1 | i | -i | i | -i | -1 | -i | -i | i | i | i | -i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | i | i | -i | -i | i | -i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -1 | i | i | -i | -i | -i | i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -i | i | i | -i | 1 | -i | -i | i | i | -i | i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -i | -i | i | i | -i | i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | i | i | -i | -i | -i | i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | i | -i | -i | i | 1 | i | i | -i | -i | i | -i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -1 | -i | -i | i | i | i | -i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | 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 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2i | -2i | 2i | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | i | 1 | -i | -1 | complex lifted from C4×S3 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2i | 2i | -2i | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | -i | 1 | i | -1 | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | -2i | 2i | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | -i | -1 | i | 1 | complex lifted from C4×S3 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2i | -2i | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | i | -1 | -i | 1 | complex lifted from C4×S3 |
| 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 | 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 11 4 8)(2 10 5 7)(3 9 6 12)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 33 22 36)(20 32 23 35)(21 31 24 34)(25 37 28 40)(26 42 29 39)(27 41 30 38)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 45)(8 44)(9 43)(10 48)(11 47)(12 46)(19 21)(22 24)(26 30)(27 29)(31 33)(34 36)(38 42)(39 41)
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), (1,11,4,8)(2,10,5,7)(3,9,6,12)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,33,22,36)(20,32,23,35)(21,31,24,34)(25,37,28,40)(26,42,29,39)(27,41,30,38), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,45)(8,44)(9,43)(10,48)(11,47)(12,46)(19,21)(22,24)(26,30)(27,29)(31,33)(34,36)(38,42)(39,41)>;
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), (1,11,4,8)(2,10,5,7)(3,9,6,12)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,33,22,36)(20,32,23,35)(21,31,24,34)(25,37,28,40)(26,42,29,39)(27,41,30,38), (1,40,16,32)(2,41,17,33)(3,42,18,34)(4,37,13,35)(5,38,14,36)(6,39,15,31)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,45)(8,44)(9,43)(10,48)(11,47)(12,46)(19,21)(22,24)(26,30)(27,29)(31,33)(34,36)(38,42)(39,41) );
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)], [(1,11,4,8),(2,10,5,7),(3,9,6,12),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,33,22,36),(20,32,23,35),(21,31,24,34),(25,37,28,40),(26,42,29,39),(27,41,30,38)], [(1,40,16,32),(2,41,17,33),(3,42,18,34),(4,37,13,35),(5,38,14,36),(6,39,15,31),(7,27,43,19),(8,28,44,20),(9,29,45,21),(10,30,46,22),(11,25,47,23),(12,26,48,24)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,45),(8,44),(9,43),(10,48),(11,47),(12,46),(19,21),(22,24),(26,30),(27,29),(31,33),(34,36),(38,42),(39,41)]])
Dic3⋊5D4 is a maximal subgroup of
Dic3⋊4D8 D4⋊S3⋊C4 D12⋊3D4 D12.D4 Dic3⋊7SD16 Q8⋊3(C4×S3) Dic3⋊SD16 D12.12D4 Dic3⋊8SD16 D24⋊9C4 D12⋊Q8 D12.Q8 Dic3⋊5D8 C24⋊C2⋊C4 D12⋊2Q8 D12.2Q8 C6.82+ 1+4 C6.2- 1+4 C6.112+ 1+4 C42⋊9D6 C42.188D6 C42.95D6 C42.97D6 C4×S3×D4 C42⋊13D6 Dic6⋊24D4 C42.114D6 C42.122D6 C4×Q8⋊3S3 C42.126D6 C42.136D6 C12⋊(C4○D4) D12⋊19D4 D12⋊20D4 C6.472+ 1+4 C4⋊C4.187D6 C4⋊C4⋊26D6 D12⋊21D4 D12⋊22D4 Dic6⋊22D4 C6.532+ 1+4 C6.202- 1+4 C6.242- 1+4 C6.1212+ 1+4 C4⋊C4⋊28D6 C6.612+ 1+4 C6.642+ 1+4 D12⋊7Q8 C42.237D6 C42.150D6 C42.151D6 C42.153D6 C42.155D6 C42⋊25D6 C42⋊26D6 C42.189D6 C42.163D6 C42.240D6 D12⋊8Q8 D12⋊9Q8 C42.178D6 D36⋊C4 C62.51C23 Dic3⋊5D12 D12⋊Dic3 C62.74C23 C62.237C23 Dic15⋊14D4 D60⋊14C4 Dic15⋊8D4 C15⋊22(C4×D4) D60⋊11C4 D60⋊3C4
Dic3⋊5D4 is a maximal quotient of
C2.(C4×Dic6) (C2×C4)⋊9D12 D6⋊C42 D6⋊C4⋊5C4 D12⋊C8 D6⋊3M4(2) C12⋊2M4(2) Dic3⋊8SD16 Dic12⋊9C4 D24⋊9C4 Dic3⋊5D8 Dic3⋊5Q16 C24⋊C2⋊C4 D24⋊10C4 D24⋊7C4 C12⋊(C4⋊C4) Dic3×C4⋊C4 (C2×D12)⋊10C4 D6⋊C4⋊6C4 D6⋊C4⋊7C4 D36⋊C4 C62.51C23 Dic3⋊5D12 D12⋊Dic3 C62.74C23 C62.237C23 Dic15⋊14D4 D60⋊14C4 Dic15⋊8D4 C15⋊22(C4×D4) D60⋊11C4 D60⋊3C4
Matrix representation of Dic3⋊5D4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 8 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 8 | 1 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,8,0,0,0,3,1],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,12,8,0,0,0,0,1] >;
Dic3⋊5D4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5D_4 % in TeX
G:=Group("Dic3:5D4"); // GroupNames label
G:=SmallGroup(96,100);
// by ID
G=gap.SmallGroup(96,100);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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