metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊4C4, Dic6⋊4C4, C12.54D4, M4(2)⋊4S3, C22.3D12, C3⋊2C4≀C2, C4.3(C4×S3), (C2×C6).1D4, C12.6(C2×C4), (C2×C4).38D6, C4○D12.2C2, (C4×Dic3)⋊1C2, C2.11(D6⋊C4), C4.29(C3⋊D4), (C3×M4(2))⋊8C2, C6.10(C22⋊C4), (C2×C12).15C22, SmallGroup(96,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊C4
G = < a,b,c | a12=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a7b >
Character table of D12⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 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 | i | -i | i | -i | -1 | 1 | -1 | -i | i | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -i | i | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | -1 | i | -i | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | -1 | i | -i | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | -1+i | 1+i | 1-i | -1-i | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | -1-i | 1-i | 1+i | -1+i | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 1+i | -1+i | -1-i | 1-i | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ20 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -2i | 2i | 1 | 1 | -1 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ21 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 2i | -2i | 1 | 1 | -1 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ22 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 1-i | -1-i | -1+i | 1+i | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T106
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 18)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18)>;
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18) );
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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)], [(2,6),(3,11),(5,9),(8,12),(13,16,19,22),(14,21,20,15),(17,24,23,18)]])
G:=TransitiveGroup(24,106);
D12⋊C4 is a maximal subgroup of
M4(2)⋊D6 D12⋊1D4 D12.4D4 D12.5D4 S3×C4≀C2 C42⋊3D6 D24⋊10C4 D24⋊7C4 M4(2)⋊24D6 C24.100D4 C24.54D4 D12⋊18D4 D12.38D4 D12.39D4 D12.40D4 Dic18⋊C4 D12⋊4Dic3 C12.80D12 C62.37D4 C60.98D4 D60⋊13C4 D60⋊10C4 D12⋊4F5 D60⋊2C4
D12⋊C4 is a maximal quotient of
C42.D6 C42.2D6 C23.35D12 C22.2D24 D12⋊2C8 Dic6⋊2C8 C12.3C42 Dic18⋊C4 D12⋊4Dic3 C12.80D12 C62.37D4 C60.98D4 D60⋊13C4 D60⋊10C4 D12⋊4F5 D60⋊2C4
Matrix representation of D12⋊C4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 1 |
| 0 | 2 | 4 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 4 | 0 | 0 | 3 |
| 0 | 1 | 2 | 0 |
| 2 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 0 | 2 | 4 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [3,0,0,1,0,2,4,0,0,4,0,0,1,0,0,0],[0,2,4,0,3,0,0,1,0,0,0,2,0,0,3,0],[2,0,0,0,0,1,2,0,0,0,4,0,4,0,0,3] >;
D12⋊C4 in GAP, Magma, Sage, TeX
D_{12}\rtimes C_4 % in TeX
G:=Group("D12:C4"); // GroupNames label
G:=SmallGroup(96,32);
// by ID
G=gap.SmallGroup(96,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,86,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^7*b>;
// generators/relations
Export
Subgroup lattice of D12⋊C4 in TeX
Character table of D12⋊C4 in TeX