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G = S4order 24 = 23·3

Symmetric group on 4 letters

non-abelian, soluble, monomial, rational

Aliases: S4, PGL2(𝔽3), SO3(𝔽3), PSO3(𝔽3), PO3(𝔽3), AGL2(𝔽2), ASL2(𝔽2), AΓL1(𝔽4), PU2(𝔽3), CSO3(𝔽3), A4⋊C2, C22⋊S3, Sym4, Σ4, Aut(Q8), Hol(C22), group of symmetries of a regular tetrahedron, group of rotations of a cube (and its dual - regular octahedron), SmallGroup(24,12)

Series: Derived Chief Lower central Upper central

C1C22A4 — S4
C1C22A4 — S4
A4 — S4
C1

Generators and relations for S4
 G = < a,b,c,d | a2=b2=c3=d2=1, cac-1=dad=ab=ba, cbc-1=a, bd=db, dcd=c-1 >

3C2
6C2
4C3
3C22
3C4
4S3
3D4

Character table of S4

 class 12A2B34
 size 13686
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3220-10    orthogonal lifted from S3
ρ43-1-101    orthogonal faithful
ρ53-110-1    orthogonal faithful

Permutation representations of S4
On 4 points: primitive, sharply 4-transitive - transitive group 4T5
Generators in S4
(1 2)(3 4)
(1 3)(2 4)
(2 3 4)
(2 4)

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

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

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

G:=TransitiveGroup(4,5);

On 6 points - transitive group 6T7
Generators in S6
(1 4)(2 5)
(2 5)(3 6)
(1 2 3)(4 5 6)
(2 3)(5 6)

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

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

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

G:=TransitiveGroup(6,7);

On 6 points - transitive group 6T8
Generators in S6
(1 4)(2 5)
(2 5)(3 6)
(1 2 3)(4 5 6)
(1 4)(2 6)(3 5)

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

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

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

G:=TransitiveGroup(6,8);

On 8 points - transitive group 8T14
Generators in S8
(1 7)(2 4)(3 5)(6 8)
(1 8)(2 5)(3 4)(6 7)
(3 4 5)(6 7 8)
(1 2)(3 7)(4 6)(5 8)

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

G:=Group( (1,7)(2,4)(3,5)(6,8), (1,8)(2,5)(3,4)(6,7), (3,4,5)(6,7,8), (1,2)(3,7)(4,6)(5,8) );

G=PermutationGroup([(1,7),(2,4),(3,5),(6,8)], [(1,8),(2,5),(3,4),(6,7)], [(3,4,5),(6,7,8)], [(1,2),(3,7),(4,6),(5,8)])

G:=TransitiveGroup(8,14);

On 12 points - transitive group 12T8
Generators in S12
(1 10)(2 5)(3 7)(4 8)(6 12)(9 11)
(1 8)(2 11)(3 6)(4 10)(5 9)(7 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(4 10)(5 12)(6 11)(7 9)

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

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

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

G:=TransitiveGroup(12,8);

On 12 points - transitive group 12T9
Generators in S12
(1 7)(3 9)(4 12)(5 10)
(1 7)(2 8)(5 10)(6 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 11)(2 10)(3 12)(4 9)(5 8)(6 7)

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

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

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

G:=TransitiveGroup(12,9);

Regular action on 24 points - transitive group 24T10
Generators in S24
(1 15)(2 12)(3 21)(4 24)(5 9)(6 16)(7 23)(8 17)(10 14)(11 19)(13 20)(18 22)
(1 19)(2 13)(3 10)(4 17)(5 22)(6 7)(8 24)(9 18)(11 15)(12 20)(14 21)(16 23)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,15)(2,12)(3,21)(4,24)(5,9)(6,16)(7,23)(8,17)(10,14)(11,19)(13,20)(18,22), (1,19)(2,13)(3,10)(4,17)(5,22)(6,7)(8,24)(9,18)(11,15)(12,20)(14,21)(16,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,15)(2,12)(3,21)(4,24)(5,9)(6,16)(7,23)(8,17)(10,14)(11,19)(13,20)(18,22), (1,19)(2,13)(3,10)(4,17)(5,22)(6,7)(8,24)(9,18)(11,15)(12,20)(14,21)(16,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,15),(2,12),(3,21),(4,24),(5,9),(6,16),(7,23),(8,17),(10,14),(11,19),(13,20),(18,22)], [(1,19),(2,13),(3,10),(4,17),(5,22),(6,7),(8,24),(9,18),(11,15),(12,20),(14,21),(16,23)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,10);

Polynomial with Galois group S4 over ℚ
actionf(x)Disc(f)
4T5x4-x+1229
6T7x6-x2-126·232
6T8x6-x4+2x2+2-213·54
8T14x8-24x6-18x5+143x4+202x3-26x2-82x-19212·54·74·174·7392
12T8x12-2x10-2x9-2x8-8x7+x6+3x5+4x4-8x3-x2-x-1-212·832·1132·2835
12T9x12-24x10+209x8-784x6+1124x4-288x2+4246·118·796

Matrix representation of S4 in GL3(ℤ) generated by

-100
010
00-1
,
100
0-10
00-1
,
010
001
100
,
-100
00-1
0-10
G:=sub<GL(3,Integers())| [-1,0,0,0,1,0,0,0,-1],[1,0,0,0,-1,0,0,0,-1],[0,0,1,1,0,0,0,1,0],[-1,0,0,0,0,-1,0,-1,0] >;

S4 in GAP, Magma, Sage, TeX

S_4
% in TeX

G:=Group("S4");
// GroupNames label

G:=SmallGroup(24,12);
// by ID

G=gap.SmallGroup(24,12);
# by ID

G:=PCGroup([4,-2,-3,-2,2,33,146,78,99,151]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^3=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,c*b*c^-1=a,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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