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## G = A6order 360 = 23·32·5

### Alternating group on 6 letters

Aliases: A6, PSL2(𝔽9), PSO-4(𝔽3), PSU2(𝔽9), Ω3(𝔽9), Ω-4(𝔽3), 3(𝔽9), PΩ-4(𝔽3), Alt(6), Alt6, also denoted L2(9) (L=PSL), SmallGroup(360,118)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — A6
 Derived series A6
 Lower central A6
 Upper central C1

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

Permutation representations of A6
On 6 points: primitive, sharply 4-transitive - transitive group 6T15
Generators in S6
```(2 3 4 5 6)
(1 6 2 3 4)```

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

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

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

`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 9 3 5 4)(2 8 6 7 10)```

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

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

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

`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 5 4 15 13)(2 7 9 3 14)(6 10 11 8 12)```

`G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5,4,15,13)(2,7,9,3,14)(6,10,11,8,12)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,5,4,15,13)(2,7,9,3,14)(6,10,11,8,12) );`

`G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,5,4,15,13),(2,7,9,3,14),(6,10,11,8,12)])`

`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 18 3 14 13)(2 8 20 16 10)(4 15 9 12 5)(6 7 19 11 17)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18,3,14,13)(2,8,20,16,10)(4,15,9,12,5)(6,7,19,11,17)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18,3,14,13)(2,8,20,16,10)(4,15,9,12,5)(6,7,19,11,17) );`

`G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,18,3,14,13),(2,8,20,16,10),(4,15,9,12,5),(6,7,19,11,17)])`

`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 23 13 28 17)(2 3 4 16 9)(5 8 29 15 21)(6 14 19 11 12)(7 18 30 26 27)(10 24 25 20 22)```

`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), (1,23,13,28,17)(2,3,4,16,9)(5,8,29,15,21)(6,14,19,11,12)(7,18,30,26,27)(10,24,25,20,22)>;`

`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), (1,23,13,28,17)(2,3,4,16,9)(5,8,29,15,21)(6,14,19,11,12)(7,18,30,26,27)(10,24,25,20,22) );`

`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)], [(1,23,13,28,17),(2,3,4,16,9),(5,8,29,15,21),(6,14,19,11,12),(7,18,30,26,27),(10,24,25,20,22)])`

`G:=TransitiveGroup(30,88);`

Polynomial with Galois group A6 over ℚ
actionf(x)Disc(f)
6T15x6-2x5-x4+2x2-126·672
10T26x10-2x9-34x8+71x7+375x6-806x5-1392x4+3042x3+568x2-1243x-11576·2836·2126814532
15T20x15+12x13+2x12+54x11+18x10+134x9+54x8+153x7+22x6+162x5-24x4+77x3-9x-1342·318·36372·119692

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`

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