metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2D4, C4⋊2D12, C12⋊1D4, C4⋊C4⋊3S3, D6⋊C4⋊8C2, C6.7(C2×D4), (C2×D12)⋊4C2, C3⋊2(C4⋊D4), (C2×C4).12D6, C2.13(S3×D4), C2.9(C2×D12), C6.34(C4○D4), (C2×C6).36C23, (C2×C12).5C22, C2.6(Q8⋊3S3), (C22×S3).7C22, C22.50(C22×S3), (C2×Dic3).31C22, (S3×C2×C4)⋊1C2, (C3×C4⋊C4)⋊6C2, SmallGroup(96,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊D4
G = < a,b,c | a12=b4=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >
Subgroups: 266 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C4⋊D4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C12⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q8⋊3S3, C12⋊D4
Character table of C12⋊D4
| 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 | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 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 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 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 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -√3 | -1 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √3 | -1 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√3 | 1 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √3 | 1 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○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 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 47 22 33)(2 42 23 28)(3 37 24 35)(4 44 13 30)(5 39 14 25)(6 46 15 32)(7 41 16 27)(8 48 17 34)(9 43 18 29)(10 38 19 36)(11 45 20 31)(12 40 21 26)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 24)(12 23)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 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), (1,47,22,33)(2,42,23,28)(3,37,24,35)(4,44,13,30)(5,39,14,25)(6,46,15,32)(7,41,16,27)(8,48,17,34)(9,43,18,29)(10,38,19,36)(11,45,20,31)(12,40,21,26), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,24)(12,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48)>;
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), (1,47,22,33)(2,42,23,28)(3,37,24,35)(4,44,13,30)(5,39,14,25)(6,46,15,32)(7,41,16,27)(8,48,17,34)(9,43,18,29)(10,38,19,36)(11,45,20,31)(12,40,21,26), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,24)(12,23)(25,29)(26,28)(30,36)(31,35)(32,34)(37,45)(38,44)(39,43)(40,42)(46,48) );
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)], [(1,47,22,33),(2,42,23,28),(3,37,24,35),(4,44,13,30),(5,39,14,25),(6,46,15,32),(7,41,16,27),(8,48,17,34),(9,43,18,29),(10,38,19,36),(11,45,20,31),(12,40,21,26)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,24),(12,23),(25,29),(26,28),(30,36),(31,35),(32,34),(37,45),(38,44),(39,43),(40,42),(46,48)]])
C12⋊D4 is a maximal subgroup of
D4⋊D12 D6⋊D8 D4⋊3D12 C3⋊C8⋊D4 Q8⋊3D12 D6⋊2SD16 Q8⋊4D12 C3⋊(C8⋊D4) D6.4SD16 C8⋊8D12 C24⋊7D4 C4.Q8⋊S3 D6.5D8 D6⋊2D8 C2.D8⋊S3 C8⋊3D12 C6.2- 1+4 C6.2+ 1+4 C6.112+ 1+4 C42⋊10D6 C42⋊11D6 C42.95D6 C42.97D6 C42.228D6 D4×D12 D4⋊5D12 C42.116D6 Q8⋊6D12 Q8⋊7D12 C42.131D6 C42.133D6 Dic6⋊20D4 S3×C4⋊D4 C6.382+ 1+4 D12⋊19D4 C4⋊C4⋊26D6 C6.172- 1+4 D12⋊21D4 Dic6⋊22D4 C6.562+ 1+4 C6.592+ 1+4 C6.1202+ 1+4 C6.1212+ 1+4 C6.662+ 1+4 C6.682+ 1+4 C42.237D6 C42.150D6 C42.153D6 C42.155D6 C42.158D6 C42⋊25D6 C42.163D6 C42⋊27D6 C42.240D6 D12⋊12D4 C42.178D6 C42.179D6 C4⋊D36 Dic3⋊D12 C12⋊7D12 Dic3⋊3D12 C12⋊2D12 C12⋊3D12 Dic5⋊D12 D30⋊D4 C60⋊6D4 C20⋊2D12 C4⋊D60
C12⋊D4 is a maximal quotient of
(C22×C4).85D6 (C2×C4)⋊9D12 D6⋊C4⋊3C4 (C2×C12)⋊5D4 C6.C22≀C2 (C2×C4).21D12 C12⋊SD16 C4⋊D24 D12.19D4 C42.36D6 Dic6⋊8D4 C4⋊Dic12 C8⋊8D12 C24⋊7D4 C8.2D12 D6⋊2D8 C8⋊3D12 D6⋊2Q16 C24.18D4 C24.19D4 C24.42D4 C4⋊C4⋊6Dic3 C4⋊(D6⋊C4) (C2×D12)⋊10C4 (C2×C4)⋊3D12 (C2×C12).56D4 C4⋊D36 Dic3⋊D12 C12⋊7D12 Dic3⋊3D12 C12⋊2D12 C12⋊3D12 Dic5⋊D12 D30⋊D4 C60⋊6D4 C20⋊2D12 C4⋊D60
Matrix representation of C12⋊D4 ►in GL4(𝔽13) generated by
| 1 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 1 | 12 |
| 3 | 7 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
G:=sub<GL(4,GF(13))| [1,1,0,0,12,0,0,0,0,0,1,1,0,0,11,12],[3,6,0,0,7,10,0,0,0,0,12,12,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,12] >;
C12⋊D4 in GAP, Magma, Sage, TeX
C_{12}\rtimes D_4 % in TeX
G:=Group("C12:D4"); // GroupNames label
G:=SmallGroup(96,102);
// by ID
G=gap.SmallGroup(96,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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