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## G = D6order 12 = 22·3

### Dihedral group

Aliases: D6, C2×S3, C6⋊C2, C3⋊C22, sometimes denoted D12 or Dih6 or Dih12, symmetries of a regular hexagon, SmallGroup(12,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — D6
 Chief series C1 — C3 — S3 — D6
 Lower central C3 — D6
 Upper central C1 — C2

Generators and relations for D6
G = < a,b | a6=b2=1, bab=a-1 >

Character table of D6

 class 1 2A 2B 2C 3 6 size 1 1 3 3 2 2 ρ1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 -1 linear of order 2 ρ4 1 -1 1 -1 1 -1 linear of order 2 ρ5 2 2 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 -1 1 orthogonal faithful

Permutation representations of D6
On 6 points - transitive group 6T3
Generators in S6
```(1 2 3 4 5 6)
(1 3)(4 6)```

`G:=sub<Sym(6)| (1,2,3,4,5,6), (1,3)(4,6)>;`

`G:=Group( (1,2,3,4,5,6), (1,3)(4,6) );`

`G=PermutationGroup([[(1,2,3,4,5,6)], [(1,3),(4,6)]])`

`G:=TransitiveGroup(6,3);`

Regular action on 12 points - transitive group 12T3
Generators in S12
```(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)```

`G:=sub<Sym(12)| (1,2,3,4,5,6)(7,8,9,10,11,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)>;`

`G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) );`

`G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]])`

`G:=TransitiveGroup(12,3);`

D6 is a maximal subgroup of   D12  C3⋊D4  GL2(𝔽3)  S5  C52⋊D6  PGL2(𝔽7)
D6 is a maximal quotient of   Dic6  D12  C3⋊D4  C52⋊D6

Polynomial with Galois group D6 over ℚ
actionf(x)Disc(f)
6T3x6-2211·36
12T3x12-3x11-15x10+30x9+85x8-73x7-168x6+73x5+121x4-36x3-25x2+5x+1218·56·376·27412

Matrix representation of D6 in GL2(ℤ) generated by

 0 -1 1 1
,
 0 -1 -1 0
`G:=sub<GL(2,Integers())| [0,1,-1,1],[0,-1,-1,0] >;`

D6 in GAP, Magma, Sage, TeX

`D_6`
`% in TeX`

`G:=Group("D6");`
`// GroupNames label`

`G:=SmallGroup(12,4);`
`// by ID`

`G=gap.SmallGroup(12,4);`
`# by ID`

`G:=PCGroup([3,-2,-2,-3,74]);`
`// Polycyclic`

`G:=Group<a,b|a^6=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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