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
| A4 — S4 |
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 >
Character table of S4
| class | 1 | 2A | 2B | 3 | 4 | |
| size | 1 | 3 | 6 | 8 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 0 | orthogonal lifted from S3 |
| ρ4 | 3 | -1 | -1 | 0 | 1 | orthogonal faithful |
| ρ5 | 3 | -1 | 1 | 0 | -1 | orthogonal faithful |
►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);
S4 is a maximal subgroup of
C42⋊S3 C22⋊S4 S5 GL3(𝔽2) A6
Cp⋊S4: C3⋊S4 C5⋊S4 C7⋊S4 C11⋊S4 C13⋊S4 C17⋊S4 C19⋊S4 ...
S4 is a maximal quotient of
CSU2(𝔽3) GL2(𝔽3) A4⋊C4 C3.S4 C42⋊S3 C22⋊S4
Cp⋊S4: C3⋊S4 C5⋊S4 C7⋊S4 C11⋊S4 C13⋊S4 C17⋊S4 C19⋊S4 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 4T5 | x4-x+1 | 229 |
| 6T7 | x6-x2-1 | 26·232 |
| 6T8 | x6-x4+2x2+2 | -213·54 |
| 8T14 | x8-24x6-18x5+143x4+202x3-26x2-82x-19 | 212·54·74·174·7392 |
| 12T8 | x12-2x10-2x9-2x8-8x7+x6+3x5+4x4-8x3-x2-x-1 | -212·832·1132·2835 |
| 12T9 | x12-24x10+209x8-784x6+1124x4-288x2+4 | 246·118·796 |
Matrix representation of S4 ►in GL3(ℤ) generated by
| -1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | -1 |
| 1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | -1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| -1 | 0 | 0 |
| 0 | 0 | -1 |
| 0 | -1 | 0 |
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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