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G = D12order 24 = 23·3

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D12, C4⋊S3, C31D4, C121C2, D61C2, C2.4D6, C6.3C22, sometimes denoted D24 or Dih12 or Dih24, SmallGroup(24,6)

Series: Derived Chief Lower central Upper central

C1C6 — D12
C1C3C6D6 — D12
C3C6 — D12
C1C2C4

Generators and relations for D12
 G = < a,b | a12=b2=1, bab=a-1 >

6C2
6C2
3C22
3C22
2S3
2S3
3D4

Character table of D12

 class 12A2B2C34612A12B
 size 116622222
ρ1111111111    trivial
ρ211-1-111111    linear of order 2
ρ3111-11-11-1-1    linear of order 2
ρ411-111-11-1-1    linear of order 2
ρ52200-12-1-1-1    orthogonal lifted from S3
ρ62200-1-2-111    orthogonal lifted from D6
ρ72-20020-200    orthogonal lifted from D4
ρ82-200-1013-3    orthogonal faithful
ρ92-200-101-33    orthogonal faithful

Permutation representations of D12
On 12 points - transitive group 12T12
Generators in S12
(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
Generators in S24
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)

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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)>;

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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22) );

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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)])

G:=TransitiveGroup(24,13);

D12 is a maximal subgroup of
C4○D12  S3×D4  Q83S3  C4⋊S4  C4.3S4  He3⋊D4  C322D12
 D12p: D24  D36  D60  D84  D132  D156  D204  D228 ...
 C6.D2p: C24⋊C2  D4⋊S3  Q82S3  C3⋊D12  C12⋊S3  C5⋊D12  C7⋊D12  C11⋊D12 ...
D12 is a maximal quotient of
C4⋊S4  C322D12
 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 ...

Polynomial with Galois group D12 over ℚ
actionf(x)Disc(f)
12T12x12-x9+8x3+1-333·76

Matrix representation of D12 in GL2(𝔽11) generated by

010
16
,
62
105
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

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