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## G = A4order 12 = 22·3

### Alternating group on 4 letters

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

 Derived series C1 — C22 — A4
 Chief series C1 — C22 — A4
 Lower central C22 — A4
 Upper central C1

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

Character table of A4

 class 1 2 3A 3B size 1 3 4 4 ρ1 1 1 1 1 trivial ρ2 1 1 ζ3 ζ32 linear of order 3 ρ3 1 1 ζ32 ζ3 linear of order 3 ρ4 3 -1 0 0 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);`

A4 is a maximal subgroup of
S4  C42⋊C3  C22⋊A4  A5  C52⋊A4  C33⋊A4
Cp⋊A4, p=1 mod 3: C7⋊A4  C13⋊A4  C19⋊A4  C31⋊A4  C37⋊A4 ...
A4 is a maximal quotient of
SL2(𝔽3)  C3.A4  C42⋊C3  C22⋊A4  C52⋊A4  C33⋊A4
Cp⋊A4, p=1 mod 3: C7⋊A4  C13⋊A4  C19⋊A4  C31⋊A4  C37⋊A4 ...

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

 0 0 1 -1 -1 -1 1 0 0
,
 0 1 0 1 0 0 -1 -1 -1
,
 1 0 0 0 0 1 -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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