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), PΩ3(𝔽4), PΩ3(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: Chief►Derived ►Lower central ►Upper central
| C1 — A5 |
Character table of A5
| class | 1 | 2 | 3 | 5A | 5B | |
| size | 1 | 15 | 20 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 3 | -1 | 0 | 1+√5/2 | 1-√5/2 | orthogonal faithful |
| ρ3 | 3 | -1 | 0 | 1-√5/2 | 1+√5/2 | orthogonal faithful |
| ρ4 | 4 | 0 | 1 | -1 | -1 | orthogonal faithful |
| ρ5 | 5 | 1 | -1 | 0 | 0 | orthogonal faithful |
►On 5 points: primitive, sharply triply transitive - transitive group 5T4
Generators in S5
(1 2 3 4 5)
(1 4 5)
G:=sub<Sym(5)| (1,2,3,4,5), (1,4,5)>;
G:=Group( (1,2,3,4,5), (1,4,5) );
G=PermutationGroup([[(1,2,3,4,5)], [(1,4,5)]])
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 9 7)(3 10 8)(4 5 6)
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,9,7)(3,10,8)(4,5,6)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,9,7)(3,10,8)(4,5,6) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,9,7),(3,10,8),(4,5,6)]])
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 11 8)(3 10 9)(5 12 6)
G:=sub<Sym(12)| (3,4,5,6,7)(8,9,10,11,12), (1,7,4)(2,11,8)(3,10,9)(5,12,6)>;
G:=Group( (3,4,5,6,7)(8,9,10,11,12), (1,7,4)(2,11,8)(3,10,9)(5,12,6) );
G=PermutationGroup([[(3,4,5,6,7),(8,9,10,11,12)], [(1,7,4),(2,11,8),(3,10,9),(5,12,6)]])
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 4 13)(2 8 15)(3 11 9)(5 7 10)(6 14 12)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,4,13)(2,8,15)(3,11,9)(5,7,10)(6,14,12)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,4,13)(2,8,15)(3,11,9)(5,7,10)(6,14,12) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,4,13),(2,8,15),(3,11,9),(5,7,10),(6,14,12)]])
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 13 5)(2 11 17)(4 20 15)(6 12 18)(7 8 16)(9 19 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), (1,13,5)(2,11,17)(4,20,15)(6,12,18)(7,8,16)(9,19,14)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,13,5)(2,11,17)(4,20,15)(6,12,18)(7,8,16)(9,19,14) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,13,5),(2,11,17),(4,20,15),(6,12,18),(7,8,16),(9,19,14)]])
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 11 28)(2 9 17)(3 26 10)(4 20 13)(5 8 6)(7 29 19)(12 24 22)(14 25 30)(15 18 21)(16 27 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), (1,11,28)(2,9,17)(3,26,10)(4,20,13)(5,8,6)(7,29,19)(12,24,22)(14,25,30)(15,18,21)(16,27,23)>;
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), (1,11,28)(2,9,17)(3,26,10)(4,20,13)(5,8,6)(7,29,19)(12,24,22)(14,25,30)(15,18,21)(16,27,23) );
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)], [(1,11,28),(2,9,17),(3,26,10),(4,20,13),(5,8,6),(7,29,19),(12,24,22),(14,25,30),(15,18,21),(16,27,23)]])
G:=TransitiveGroup(30,9);
A5 is a maximal subgroup of
S5 A6
A5 is a maximal quotient of SL2(𝔽5)
| action | f(x) | Disc(f) |
|---|---|---|
| 5T4 | x5-x2-2x-3 | 172·292 |
| 6T12 | x6-x5-x4-x3-2x2-2 | 24·5772 |
| 10T7 | x10-4x9-9x8+51x7-x6-188x5+125x4+207x3-189x2-19x+19 | 194·2934·24112 |
| 12T33 | x12-4x11+2x10+19x8-12x7-4x6-12x5+19x4+2x2-4x+1 | 242·32·178 |
| 15T5 | x15-3x14-12x13+24x12+72x11-39x10-121x9-282x8-201x7+276x6+894x5+1026x4+687x3+288x2+72x+9 | 210·324·510·134·2310·294·312 |
Matrix representation of A5 ►in GL3(𝔽5) generated by
| 2 | 4 | 0 |
| 4 | 4 | 3 |
| 2 | 2 | 2 |
| 0 | 4 | 2 |
| 2 | 0 | 1 |
| 3 | 1 | 0 |
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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