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G = A4order 12 = 22·3

Alternating group on 4 letters

metabelian, soluble, monomial, A-group

Aliases: A4, PSL2(𝔽3), AGL1(𝔽4), PSU2(𝔽3), Ω3(𝔽3), 3(𝔽3), C22⋊C3, Alt4, also denoted L2(3) (L=PSL), group of rotations of a regular tetrahedron, SmallGroup(12,3)

Series: Derived Chief Lower central Upper central

C1C22 — A4
C1C22 — A4
C22 — A4
C1

Generators and relations for A4
 G = < a,b,c | a2=b2=c3=1, cac-1=ab=ba, cbc-1=a >

3C2
4C3

Character table of A4

 class 123A3B
 size 1344
ρ11111    trivial
ρ211ζ3ζ32    linear of order 3
ρ311ζ32ζ3    linear of order 3
ρ43-100    orthogonal faithful

Permutation representations of A4
On 4 points: primitive, sharply doubly transitive - transitive group 4T4
Generators in S4
(1 4)(2 3)
(1 2)(3 4)
(2 3 4)

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

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

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

G:=TransitiveGroup(4,4);

On 6 points - transitive group 6T4
Generators in S6
(2 5)(3 6)
(1 4)(3 6)
(1 2 3)(4 5 6)

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

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

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

G:=TransitiveGroup(6,4);

Regular action on 12 points - transitive group 12T4
Generators in S12
(1 4)(2 9)(3 12)(5 11)(6 7)(8 10)
(1 10)(2 5)(3 7)(4 8)(6 12)(9 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,4);

Polynomial with Galois group A4 over ℚ
actionf(x)Disc(f)
4T4x4+2x3+2x2+226·72
6T4x6+x4-2x2-126·74
12T4x12+4x10+24x8+48x6-560x4+3136290·714·43394

Matrix representation of A4 in GL3(ℤ) generated by

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

A4 in GAP, Magma, Sage, TeX

A_4
% in TeX

G:=Group("A4");
// GroupNames label

G:=SmallGroup(12,3);
// by ID

G=gap.SmallGroup(12,3);
# by ID

G:=PCGroup([3,-3,-2,2,37,83]);
// Polycyclic

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

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