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G = D6order 12 = 22·3

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, rational, 2-hyperelementary

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

C1C3 — D6
C1C3S3 — D6
C3 — D6
C1C2

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

3C2
3C2
3C22

Character table of D6

 class 12A2B2C36
 size 113322
ρ1111111    trivial
ρ211-1-111    linear of order 2
ρ31-1-111-1    linear of order 2
ρ41-11-11-1    linear of order 2
ρ52200-1-1    orthogonal lifted from S3
ρ62-200-11    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);

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
11
,
0-1
-10
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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