metabelian, soluble, monomial, A-group
Aliases: A4, PSL2(𝔽3), AGL1(𝔽4), PSU2(𝔽3), Ω3(𝔽3), PΩ3(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
| C22 — A4 |
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 |
►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 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 4T4 | x4+2x3+2x2+2 | 26·72 |
| 6T4 | x6+x4-2x2-1 | 26·74 |
| 12T4 | x12+4x10+24x8+48x6-560x4+3136 | 290·714·43394 |
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
Export