non-abelian, simple, perfect, not soluble
Aliases: A6, PSL2(𝔽9), PSO-4(𝔽3), PSU2(𝔽9), Ω3(𝔽9), Ω-4(𝔽3), PΩ3(𝔽9), PΩ-4(𝔽3), Alt(6), Alt6, also denoted L2(9) (L=PSL), SmallGroup(360,118)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — A6 |
45C2
20C3
20C3
36C5
15C22
15C22
45C4
60S3
60S3
10C32
36D5
45D4
15A4
15A4
10C3⋊S3
15S4
15S4
10C32⋊C4
6A5
6A5
Character table of A6
| class | 1 | 2 | 3A | 3B | 4 | 5A | 5B | |
| size | 1 | 45 | 40 | 40 | 90 | 72 | 72 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 5 | 1 | 2 | -1 | -1 | 0 | 0 | orthogonal faithful |
| ρ3 | 5 | 1 | -1 | 2 | -1 | 0 | 0 | orthogonal faithful |
| ρ4 | 8 | 0 | -1 | -1 | 0 | 1-√5/2 | 1+√5/2 | orthogonal faithful |
| ρ5 | 8 | 0 | -1 | -1 | 0 | 1+√5/2 | 1-√5/2 | orthogonal faithful |
| ρ6 | 9 | 1 | 0 | 0 | 1 | -1 | -1 | orthogonal faithful |
| ρ7 | 10 | -2 | 1 | 1 | 0 | 0 | 0 | orthogonal faithful |
►On 6 points: primitive, sharply 4-transitive - transitive group 6T15
Generators in S6
(2 3 4 5 6)
(1 3 4 5 6)
G:=sub<Sym(6)| (2,3,4,5,6), (1,3,4,5,6)>;
G:=Group( (2,3,4,5,6), (1,3,4,5,6) );
G=PermutationGroup([[(2,3,4,5,6)], [(1,3,4,5,6)]])
G:=TransitiveGroup(6,15);
►On 10 points: primitive, doubly transitive - transitive group 10T26
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 4 5 3 7)(2 8 10 9 6)
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,4,5,3,7)(2,8,10,9,6)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,4,5,3,7)(2,8,10,9,6) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,4,5,3,7),(2,8,10,9,6)]])
G:=TransitiveGroup(10,26);
►On 15 points: primitive - transitive group 15T20
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 7 6 5 9)(2 15 14 13 4)(3 11 8 10 12)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,7,6,5,9)(2,15,14,13,4)(3,11,8,10,12)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,7,6,5,9)(2,15,14,13,4)(3,11,8,10,12) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,7,6,5,9),(2,15,14,13,4),(3,11,8,10,12)]])
G:=TransitiveGroup(15,20);
►On 20 points - transitive group 20T89
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16 17 3 8)(2 13 6 10 11)(4 5 20 12 18)(7 19 9 15 14)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,17,3,8)(2,13,6,10,11)(4,5,20,12,18)(7,19,9,15,14)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,17,3,8)(2,13,6,10,11)(4,5,20,12,18)(7,19,9,15,14) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16,17,3,8),(2,13,6,10,11),(4,5,20,12,18),(7,19,9,15,14)]])
G:=TransitiveGroup(20,89);
►On 30 points - transitive group 30T88
Generators in S30
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 2 18 13 5)(3 12 28 21 8)(4 10 24 27 19)(6 7 17 9 14)(11 20 29 30 26)(15 25 16 22 23)
G:=sub<Sym(30)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,2,18,13,5)(3,12,28,21,8)(4,10,24,27,19)(6,7,17,9,14)(11,20,29,30,26)(15,25,16,22,23)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,2,18,13,5)(3,12,28,21,8)(4,10,24,27,19)(6,7,17,9,14)(11,20,29,30,26)(15,25,16,22,23) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,2,18,13,5),(3,12,28,21,8),(4,10,24,27,19),(6,7,17,9,14),(11,20,29,30,26),(15,25,16,22,23)]])
G:=TransitiveGroup(30,88);
Polynomial with Galois group A6 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 6T15 | x6-2x5-x4+2x2-1 | 26·672 |
| 10T26 | x10-2x9-34x8+71x7+375x6-806x5-1392x4+3042x3+568x2-1243x-115 | 76·2836·2126814532 |
| 15T20 | x15+12x13+2x12+54x11+18x10+134x9+54x8+153x7+22x6+162x5-24x4+77x3-9x-1 | 342·318·36372·119692 |
Matrix representation of A6 ►in GL4(𝔽2) generated by
| 1 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 |
G:=sub<GL(4,GF(2))| [1,1,0,0,1,1,0,1,0,0,1,1,1,0,1,0],[1,1,1,1,0,1,1,1,1,1,0,0,1,1,0,1] >;
A6 in GAP, Magma, Sage, TeX
A_6
% in TeX
G:=Group("A6"); // GroupNames label
G:=SmallGroup(360,118);
// by ID
G=gap.SmallGroup(360,118);
# by ID
Export