metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊1D12, C12⋊4D4, C42⋊6S3, (C4×C12)⋊4C2, C6.3(C2×D4), (C2×D12)⋊1C2, C3⋊1(C4⋊1D4), (C2×C4).76D6, C2.5(C2×D12), (C2×C6).15C23, (C2×C12).87C22, (C22×S3).1C22, C22.36(C22×S3), SmallGroup(96,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D12
G = < a,b,c | a4=b12=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 330 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C42, C2×D4, D12, C2×C12, C22×S3, C4⋊1D4, C4×C12, C2×D12, C4⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4⋊1D4, C2×D12, C4⋊D12
Character table of C4⋊D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | -2 | 1 | 1 | -1 | √3 | -√3 | 1 | √3 | -√3 | 1 | -√3 | √3 | -1 | √3 | -√3 | -1 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | √3 | -√3 | -√3 | -1 | -1 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | √3 | -√3 | √3 | -√3 | -√3 | √3 | -√3 | 1 | 1 | √3 | orthogonal lifted from D12 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 2 | 1 | 1 | -1 | -√3 | √3 | -1 | √3 | -√3 | -1 | -√3 | √3 | 1 | -√3 | √3 | 1 | orthogonal lifted from D12 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 2 | -2 | 0 | -1 | 1 | 1 | √3 | -√3 | -√3 | -1 | -1 | √3 | 1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 2 | -2 | 0 | -1 | 1 | 1 | -√3 | √3 | √3 | -1 | -1 | -√3 | 1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -2 | 2 | 0 | -1 | 1 | 1 | -√3 | √3 | -√3 | 1 | 1 | √3 | -1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 2 | 1 | 1 | -1 | √3 | -√3 | -1 | -√3 | √3 | -1 | √3 | -√3 | 1 | √3 | -√3 | 1 | orthogonal lifted from D12 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -2 | 2 | 0 | -1 | 1 | 1 | √3 | -√3 | √3 | 1 | 1 | -√3 | -1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | -√3 | √3 | √3 | -1 | -1 | -√3 | orthogonal lifted from D12 |
| ρ29 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | -2 | 1 | 1 | -1 | -√3 | √3 | 1 | -√3 | √3 | 1 | √3 | -√3 | -1 | -√3 | √3 | -1 | orthogonal lifted from D12 |
| ρ30 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | -√3 | √3 | -√3 | √3 | √3 | -√3 | √3 | 1 | 1 | -√3 | orthogonal lifted from D12 |
►On 48 points
(1 25 19 46)(2 26 20 47)(3 27 21 48)(4 28 22 37)(5 29 23 38)(6 30 24 39)(7 31 13 40)(8 32 14 41)(9 33 15 42)(10 34 16 43)(11 35 17 44)(12 36 18 45)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)
G:=sub<Sym(48)| (1,25,19,46)(2,26,20,47)(3,27,21,48)(4,28,22,37)(5,29,23,38)(6,30,24,39)(7,31,13,40)(8,32,14,41)(9,33,15,42)(10,34,16,43)(11,35,17,44)(12,36,18,45), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;
G:=Group( (1,25,19,46)(2,26,20,47)(3,27,21,48)(4,28,22,37)(5,29,23,38)(6,30,24,39)(7,31,13,40)(8,32,14,41)(9,33,15,42)(10,34,16,43)(11,35,17,44)(12,36,18,45), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );
G=PermutationGroup([[(1,25,19,46),(2,26,20,47),(3,27,21,48),(4,28,22,37),(5,29,23,38),(6,30,24,39),(7,31,13,40),(8,32,14,41),(9,33,15,42),(10,34,16,43),(11,35,17,44),(12,36,18,45)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,15),(16,24),(17,23),(18,22),(19,21),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])
C4⋊D12 is a maximal subgroup of
C4.D24 C8⋊5D12 C4.5D24 C12⋊4D8 C8⋊D12 C42.19D6 Q8⋊5D12 C4⋊D24 Dic6⋊8D4 C12⋊7D8 Q8⋊2D12 C42.64D6 C42.70D6 C12⋊D8 C12⋊6SD16 C12.D8 C42.276D6 C42⋊11D6 C42.100D6 D4×D12 Dic6⋊24D4 Q8⋊7D12 C42.136D6 C42⋊20D6 C42.155D6 C42⋊27D6 S3×C4⋊1D4 C42.240D6 C42⋊6D9 (C4×C12)⋊C6 C12⋊D12 C12⋊4D12 C20⋊D12 C42⋊6D15
C4⋊D12 is a maximal quotient of
(C2×C12)⋊5D4 (C2×C12).33D4 C8⋊5D12 C12⋊4D8 C8.8D12 C12⋊4Q16 C8⋊D12 C8.D12 C42⋊10Dic3 (C2×C4)⋊6D12 C42⋊6D9 C12⋊D12 C12⋊4D12 C20⋊D12 C42⋊6D15
Matrix representation of C4⋊D12 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 10 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,7,0,0,6,10],[10,10,0,0,3,7,0,0,0,0,12,1,0,0,12,0],[1,0,0,0,1,12,0,0,0,0,12,0,0,0,12,1] >;
C4⋊D12 in GAP, Magma, Sage, TeX
C_4\rtimes D_{12} % in TeX
G:=Group("C4:D12"); // GroupNames label
G:=SmallGroup(96,81);
// by ID
G=gap.SmallGroup(96,81);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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