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
| C3 — D6 |
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 |
►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
| action | f(x) | Disc(f) |
|---|---|---|
| 6T3 | x6-2 | 211·36 |
| 12T3 | x12-3x11-15x10+30x9+85x8-73x7-168x6+73x5+121x4-36x3-25x2+5x+1 | 218·56·376·27412 |
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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