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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 3)

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

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

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

G:=TransitiveGroup(5,4);

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

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

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

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

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 4 6)(2 8 10)(3 7 9)(5 12 11)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(30,9);

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

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