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G = A6order 360 = 23·32·5

Alternating group on 6 letters

non-abelian, simple, perfect, not soluble

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

C1 — A6
A6
A6
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 123A3B45A5B
 size 1454040907272
ρ11111111    trivial
ρ2512-1-100    orthogonal faithful
ρ351-12-100    orthogonal faithful
ρ480-1-101-5/21+5/2    orthogonal faithful
ρ580-1-101+5/21-5/2    orthogonal faithful
ρ691001-1-1    orthogonal faithful
ρ710-211000    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

1101
1100
0011
0110
,
1011
1111
1100
1101
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

Subgroup lattice of A6 in TeX
Character table of A6 in TeX

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