Copied to
clipboard

## G = D12order 24 = 23·3

### Dihedral group

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

 Derived series C1 — C6 — D12
 Chief series C1 — C3 — C6 — D6 — D12
 Lower central C3 — C6 — D12
 Upper central C1 — C2 — C4

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

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 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  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

 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

׿
×
𝔽