Copied to
clipboard

G = S3order 6 = 2·3

Symmetric group on 3 letters

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), 3(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

C1C3 — S3
C1C3 — S3
C3 — S3
C1

Generators and relations for S3
 G = < a,b | a3=b2=1, bab=a-1 >

3C2

Character table of S3

 class 123
 size 132
ρ1111    trivial
ρ21-11    linear of order 2
ρ320-1    orthogonal faithful

Permutation representations of S3
On 3 points: primitive, sharply triply transitive - transitive group 3T2
Generators in S3
(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
Generators in S6
(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);

Polynomial with Galois group S3 over ℚ
actionf(x)Disc(f)
3T2x3-2-22·33
6T2x6-3x5-2x4+9x3-5x+124·373

Matrix representation of S3 in GL2(ℤ) generated by

01
-1-1
,
10
-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

׿
×
𝔽