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G = A5order 60 = 22·3·5

Alternating group on 5 letters

non-abelian, simple, perfect, not soluble, A-group

Aliases: A5, SL2(𝔽4), PGL2(𝔽4), PSL2(𝔽5), PSL2(𝔽4), SO3(𝔽4), SU2(𝔽4), O3(𝔽4), PSO3(𝔽4), PO3(𝔽4), PU2(𝔽4), PSU2(𝔽5), PSU2(𝔽4), CSU2(𝔽4), Spin3(𝔽4), Ω3(𝔽5), Ω3(𝔽4), Ω-4(𝔽2), 3(𝔽4), 3(𝔽5), PΩ-4(𝔽2), Alt(5), Alt5, also denoted L2(5) (L=PSL), also denoted L2(4) (L=PSL), group of rotations of a regular icosahedron (and its dual dodecahedron), SmallGroup(60,5)

Series: ChiefDerived Lower central Upper central

C1 — A5
A5
A5
C1

15C2
10C3
6C5
5C22
10S3
6D5
5A4

Character table of A5

 class 1235A5B
 size 115201212
ρ111111    trivial
ρ23-101+5/21-5/2    orthogonal faithful
ρ33-101-5/21+5/2    orthogonal faithful
ρ4401-1-1    orthogonal faithful
ρ551-100    orthogonal faithful

Permutation representations of A5
On 5 points: primitive, sharply triply transitive - transitive group 5T4
Generators in S5
(1 2 3 4 5)
(1 2 5)

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

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

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

G:=TransitiveGroup(5,4);

On 6 points: primitive, doubly transitive - transitive group 6T12
Generators in S6
(2 3 4 5 6)
(1 3 5)(2 6 4)

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

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

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

G:=TransitiveGroup(6,12);

On 10 points: primitive - transitive group 10T7
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 2 6)(3 9 7)(5 10 8)

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

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

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

G:=TransitiveGroup(10,7);

On 12 points - transitive group 12T33
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 7 4)(2 9 11)(3 8 12)(5 10 6)

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

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

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

G:=TransitiveGroup(12,33);

On 15 points - transitive group 15T5
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 13 11)(2 5 9)(3 14 6)(4 7 15)(8 12 10)

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

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

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

G:=TransitiveGroup(15,5);

On 20 points - transitive group 20T15
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 14 19)(2 10 20)(3 17 8)(4 18 11)(5 6 7)(12 13 16)

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

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

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

G:=TransitiveGroup(20,15);

On 30 points - transitive group 30T9
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 25 23)(2 15 6)(3 21 27)(4 9 22)(5 30 12)(7 29 24)(8 13 17)(10 20 26)(11 16 19)(14 28 18)

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

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

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

G:=TransitiveGroup(30,9);

A5 is a maximal subgroup of   S5  A6
A5 is a maximal quotient of   SL2(𝔽5)

Polynomial with Galois group A5 over ℚ
actionf(x)Disc(f)
5T4x5-x2-2x-3172·292
6T12x6-x5-x4-x3-2x2-224·5772
10T7x10-4x9-9x8+51x7-x6-188x5+125x4+207x3-189x2-19x+19194·2934·24112
12T33x12-4x11+2x10+19x8-12x7-4x6-12x5+19x4+2x2-4x+1242·32·178
15T5x15-3x14-12x13+24x12+72x11-39x10-121x9-282x8-201x7+276x6+894x5+1026x4+687x3+288x2+72x+9210·324·510·134·2310·294·312

Matrix representation of A5 in GL3(𝔽5) generated by

240
443
222
,
042
201
310
G:=sub<GL(3,GF(5))| [2,4,2,4,4,2,0,3,2],[0,2,3,4,0,1,2,1,0] >;

A5 in GAP, Magma, Sage, TeX

A_5
% in TeX

G:=Group("A5");
// GroupNames label

G:=SmallGroup(60,5);
// by ID

G=gap.SmallGroup(60,5);
# by ID

Export

Subgroup lattice of A5 in TeX
Character table of A5 in TeX

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