metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊3D4, Dic3⋊1D4, C23.15D6, (C2×D4)⋊6S3, (C6×D4)⋊4C2, (C2×D12)⋊9C2, C3⋊2(C4⋊1D4), C4⋊1(C3⋊D4), (C2×C4).52D6, C6.52(C2×D4), C2.28(S3×D4), (C4×Dic3)⋊6C2, (C2×C6).55C23, (C2×C12).35C22, C22.62(C22×S3), (C22×C6).22C22, (C22×S3).12C22, (C2×Dic3).39C22, (C2×C3⋊D4)⋊7C2, C2.16(C2×C3⋊D4), SmallGroup(96,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊3D4
G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 290 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×D4, C2×D4, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊1D4, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, C12⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, S3×D4, C2×C3⋊D4, C12⋊3D4
Character table of C12⋊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 | 12 | 12 | 2 | 2 | 2 | 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 | 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 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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 D6 |
| ρ14 | 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 |
| 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 S3 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 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 |
| ρ18 | 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 |
| 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 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 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 |
| 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 S3×D4 |
| 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 | orthogonal lifted from S3×D4 |
►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 47 34 24)(2 40 35 17)(3 45 36 22)(4 38 25 15)(5 43 26 20)(6 48 27 13)(7 41 28 18)(8 46 29 23)(9 39 30 16)(10 44 31 21)(11 37 32 14)(12 42 33 19)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 31)(26 30)(27 29)(32 36)(33 35)
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), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35)>;
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), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35) );
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)], [(1,47,34,24),(2,40,35,17),(3,45,36,22),(4,38,25,15),(5,43,26,20),(6,48,27,13),(7,41,28,18),(8,46,29,23),(9,39,30,16),(10,44,31,21),(11,37,32,14),(12,42,33,19)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47),(25,31),(26,30),(27,29),(32,36),(33,35)]])
C12⋊3D4 is a maximal subgroup of
C23.2D12 D12⋊1D4 Dic3.SD16 Dic6⋊2D4 C4⋊C4.D6 D12⋊3D4 Dic3⋊D8 C24⋊5D4 C24⋊11D4 Dic3⋊5SD16 C24⋊15D4 C24⋊9D4 D12⋊18D4 2+ 1+4⋊6S3 C42.228D6 Dic6⋊24D4 C42.114D6 C42.116D6 C24⋊8D6 C24.45D6 C24.47D6 C12⋊(C4○D4) Dic6⋊20D4 C6.382+ 1+4 D12⋊19D4 C6.442+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C6.682+ 1+4 C42.233D6 C42⋊20D6 C42.143D6 S3×C4⋊1D4 C42⋊28D6 Dic6⋊11D4 D4×C3⋊D4 C24.52D6 C6.1462+ 1+4 (C2×C12)⋊17D4 C6.1482+ 1+4 C36⋊D4 C12⋊D12 C62.84C23 C62.121C23 C62.258C23 C12⋊D20 C60⋊10D4 Dic15⋊5D4 C60⋊3D4
C12⋊3D4 is a maximal quotient of
C24.18D6 C24.24D6 C23⋊3D12 (C4×Dic3)⋊8C4 (C2×D12)⋊10C4 (C2×C12).290D4 C42.64D6 C42.214D6 C42.65D6 C12⋊D8 C42.74D6 C12⋊4SD16 C12⋊6SD16 C42.80D6 C12⋊3Q16 C24⋊5D4 C24⋊11D4 C24.22D4 C24.31D4 C24.43D4 C24⋊15D4 C24⋊9D4 C24.26D4 C24.37D4 C24.28D4 C24.30D6 C24.32D6 C36⋊D4 C12⋊D12 C62.84C23 C62.121C23 C62.258C23 C12⋊D20 C60⋊10D4 Dic15⋊5D4 C60⋊3D4
Matrix representation of C12⋊3D4 ►in GL4(𝔽13) generated by
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 2 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,0,0,0,0,0,4,2,0,0,11,9],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C12⋊3D4 in GAP, Magma, Sage, TeX
C_{12}\rtimes_3D_4 % in TeX
G:=Group("C12:3D4"); // GroupNames label
G:=SmallGroup(96,147);
// by ID
G=gap.SmallGroup(96,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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