metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D12, C4⋊S3, C3⋊1D4, C12⋊1C2, D6⋊1C2, C2.4D6, C6.3C22, sometimes denoted D24 or Dih12 or Dih24, SmallGroup(24,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12
G = < a,b | a12=b2=1, bab=a-1 >
Character table of D12
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √3 | -√3 | orthogonal faithful |
| ρ9 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√3 | √3 | orthogonal faithful |
►On 12 points - transitive group 12T12
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 3)(4 12)(5 11)(6 10)(7 9)
G:=sub<Sym(12)| (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12)], [(1,3),(4,12),(5,11),(6,10),(7,9)]])
G:=TransitiveGroup(12,12);
►Regular action on 24 points - transitive group 24T13
(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)
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)>;
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) );
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)]])
G:=TransitiveGroup(24,13);
D12 is a maximal subgroup of
C4○D12 S3×D4 Q8⋊3S3 C4⋊S4 C4.3S4 He3⋊D4 C32⋊2D12
D12p: D24 D36 D60 D84 D132 D156 D204 D228 ...
C6.D2p: C24⋊C2 D4⋊S3 Q8⋊2S3 C3⋊D12 C12⋊S3 C5⋊D12 C7⋊D12 C11⋊D12 ...
D12 is a maximal quotient of
C4⋊S4 C32⋊2D12
D12p: D24 D36 D60 D84 D132 D156 D204 D228 ...
C6.D2p: C24⋊C2 Dic12 C4⋊Dic3 D6⋊C4 C3⋊D12 C12⋊S3 C5⋊D12 C7⋊D12 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 12T12 | x12-x9+8x3+1 | -333·76 |
Matrix representation of D12 ►in GL2(𝔽11) generated by
| 0 | 10 |
| 1 | 6 |
| 6 | 2 |
| 10 | 5 |
G:=sub<GL(2,GF(11))| [0,1,10,6],[6,10,2,5] >;
D12 in GAP, Magma, Sage, TeX
D_{12} % in TeX
G:=Group("D12"); // GroupNames label
G:=SmallGroup(24,6);
// by ID
G=gap.SmallGroup(24,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-3,49,21,259]);
// Polycyclic
G:=Group<a,b|a^12=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D12 in TeX
Character table of D12 in TeX