metacyclic, supersoluble, monomial, Z-group, rational, 2-hyperelementary
Aliases: S3, D3, GL2(𝔽2), SL2(𝔽2), PGL2(𝔽2), PSL2(𝔽2), SO3(𝔽2), SU2(𝔽2), O3(𝔽2), PSO3(𝔽2), PO3(𝔽2), AGL1(𝔽3), PU2(𝔽2), PSU2(𝔽2), CO3(𝔽2), CSU2(𝔽2), CSO3(𝔽2), Spin3(𝔽2), Ω3(𝔽2), PΩ3(𝔽2), C3⋊C2, sometimes denoted D6 or Dih3 or Dih6, Sym3, Σ3, symmetries of a regular triangle, 1st non-abelian group, SmallGroup(6,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3 |
Generators and relations for S3
G = < a,b | a3=b2=1, bab=a-1 >
Character table of S3
| class | 1 | 2 | 3 | |
| size | 1 | 3 | 2 | |
| ρ1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 2 | 0 | -1 | orthogonal faithful |
►On 3 points: primitive, sharply triply transitive - transitive group 3T2
(1 2 3)
(2 3)
G:=sub<Sym(3)| (1,2,3), (2,3)>;
G:=Group( (1,2,3), (2,3) );
G=PermutationGroup([[(1,2,3)], [(2,3)]])
G:=TransitiveGroup(3,2);
►Regular action on 6 points - transitive group 6T2
(1 2 3)(4 5 6)
(1 4)(2 6)(3 5)
G:=sub<Sym(6)| (1,2,3)(4,5,6), (1,4)(2,6)(3,5)>;
G:=Group( (1,2,3)(4,5,6), (1,4)(2,6)(3,5) );
G=PermutationGroup([[(1,2,3),(4,5,6)], [(1,4),(2,6),(3,5)]])
G:=TransitiveGroup(6,2);
S3 is a maximal subgroup of
C3⋊S3 S4 A5 C52⋊S3 C72⋊S3
D3p: D9 D15 D21 D33 D39 D51 D57 D69 ...
S3 is a maximal quotient of
Dic3 C3⋊S3 S4 C52⋊S3 C72⋊S3 ΓL2(𝔽4)
D3p: D9 D15 D21 D33 D39 D51 D57 D69 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 3T2 | x3-2 | -22·33 |
| 6T2 | x6-3x5-2x4+9x3-5x+1 | 24·373 |
Matrix representation of S3 ►in GL2(ℤ) generated by
| 0 | 1 |
| -1 | -1 |
| 1 | 0 |
| -1 | -1 |
G:=sub<GL(2,Integers())| [0,-1,1,-1],[1,-1,0,-1] >;
S3 in GAP, Magma, Sage, TeX
S_3
% in TeX
G:=Group("S3"); // GroupNames label
G:=SmallGroup(6,1);
// by ID
G=gap.SmallGroup(6,1);
# by ID
G:=PCGroup([2,-2,-3,17]);
// Polycyclic
G:=Group<a,b|a^3=b^2=1,b*a*b=a^-1>;
// generators/relations
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