metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.35D4, D4.10D12, Q8.15D12, C42.26D6, Dic6.35D4, M4(2).8D6, C4≀C2⋊4S3, C8⋊D6⋊9C2, Q8○D12⋊1C2, (C3×D4).5D4, C12.6(C2×D4), (C3×Q8).5D4, D4⋊D6⋊2C2, C4○D4.20D6, C4.128(S3×D4), C4.12(C2×D12), C6.30C22≀C2, C3⋊2(D4.8D4), (C2×Dic3).3D4, C42⋊4S3⋊10C2, C22.32(S3×D4), C42⋊7S3⋊11C2, C12.47D4⋊2C2, (C4×C12).53C22, C2.33(D6⋊D4), (C2×C12).267C23, C4○D12.16C22, (C2×D12).71C22, (C2×Dic6).77C22, (C3×M4(2)).5C22, C4.Dic3.11C22, (C3×C4≀C2)⋊4C2, (C2×C6).29(C2×D4), (C3×C4○D4).8C22, (C2×C4).112(C22×S3), SmallGroup(192,386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.35D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a9c-1 >
Subgroups: 480 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6, C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×4], C4×S3 [×3], D12, D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C24⋊C2, D24, C4.Dic3, D6⋊C4 [×2], D4⋊S3, Q8⋊2S3, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C2×D12, C4○D12, C4○D12, D4⋊2S3 [×3], S3×Q8, C3×C4○D4, D4.8D4, C42⋊4S3, C12.47D4, C3×C4≀C2, C42⋊7S3, C8⋊D6, D4⋊D6, Q8○D12, D12.35D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D4.8D4, D6⋊D4, D12.35D4
Character table of D12.35D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 2 | 4 | 8 | 8 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | |
| 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 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 1 | 2 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | 1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | -1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | 1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | -1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | -2i | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 2i | 0 | 0 | 0 | complex lifted from D4.8D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | ζ4+2ζ32+1 | 0 | ζ43+2ζ32+1 | ζ43+2ζ3+1 | ζ4+2ζ3+1 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | ζ43+2ζ3+1 | 0 | ζ4+2ζ3+1 | ζ4+2ζ32+1 | ζ43+2ζ32+1 | 0 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | ζ4+2ζ3+1 | 0 | ζ43+2ζ3+1 | ζ43+2ζ32+1 | ζ4+2ζ32+1 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | ζ43+2ζ32+1 | 0 | ζ4+2ζ32+1 | ζ4+2ζ3+1 | ζ43+2ζ3+1 | 0 | 0 | 0 | complex faithful |
►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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 25 22 31)(17 26 23 32)(18 27 24 33)
(1 34 7 28)(2 35 8 29)(3 36 9 30)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,34,19,28)(14,35,20,29)(15,36,21,30)(16,25,22,31)(17,26,23,32)(18,27,24,33), (1,34,7,28)(2,35,8,29)(3,36,9,30)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,34,19,28)(14,35,20,29)(15,36,21,30)(16,25,22,31)(17,26,23,32)(18,27,24,33), (1,34,7,28)(2,35,8,29)(3,36,9,30)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,34,19,28),(14,35,20,29),(15,36,21,30),(16,25,22,31),(17,26,23,32),(18,27,24,33)], [(1,34,7,28),(2,35,8,29),(3,36,9,30),(4,25,10,31),(5,26,11,32),(6,27,12,33),(13,40,19,46),(14,41,20,47),(15,42,21,48),(16,43,22,37),(17,44,23,38),(18,45,24,39)])
Matrix representation of D12.35D4 ►in GL6(𝔽73)
| 63 | 52 | 0 | 0 | 0 | 0 |
| 47 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 1 |
| 0 | 0 | 2 | 1 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 63 | 1 | 0 | 0 | 0 | 0 |
| 47 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 72 | 1 | 1 |
| 72 | 71 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 19 | 46 | 27 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 46 | 27 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 19 | 46 | 27 | 27 |
G:=sub<GL(6,GF(73))| [63,47,0,0,0,0,52,11,0,0,0,0,0,0,72,2,0,0,0,0,72,1,0,0,0,0,0,72,0,72,0,0,1,72,1,0],[63,47,0,0,0,0,1,10,0,0,0,0,0,0,72,0,0,71,0,0,0,0,1,72,0,0,0,1,0,1,0,0,0,0,0,1],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,27,19,0,0,0,0,27,46,0,0,0,0,0,27,27,0,0,0,0,27,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,19,0,0,46,0,46,46,0,0,27,46,0,27,0,0,0,0,0,27] >;
D12.35D4 in GAP, Magma, Sage, TeX
D_{12}._{35}D_4 % in TeX
G:=Group("D12.35D4"); // GroupNames label
G:=SmallGroup(192,386);
// by ID
G=gap.SmallGroup(192,386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,136,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^9*c^-1>;
// generators/relations