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G = S5order 120 = 23·3·5

Symmetric group on 5 letters

non-abelian, almost simple, not soluble, rational

Aliases: S5, PGL2(𝔽5), SO3(𝔽5), SO-4(𝔽2), O-4(𝔽2), PSO3(𝔽5), PSO-4(𝔽2), PO3(𝔽5), PO-4(𝔽2), PU2(𝔽5), PΓL2(𝔽4), PΣL2(𝔽4), PΓU2(𝔽4), PΣU2(𝔽4), CO-4(𝔽2), CSO3(𝔽5), CSO-4(𝔽2), PCO-4(𝔽2), PCSO-4(𝔽2), ΣL2(𝔽4), A5⋊C2, Sym(5), Sym5, Σ5, Aut(A5), group of symmetries of a 5-cell (4-simplex), SmallGroup(120,34)

Series: ChiefDerived Lower central Upper central

C1A5 — S5
A5 — S5
A5 — S5
C1

10C2
15C2
10C3
6C5
5C22
15C22
15C4
10C6
10S3
10S3
6D5
15D4
5A4
10D6
6F5
5S4

Character table of S5

 class 12A2B3456
 size 1101520302420
ρ11111111    trivial
ρ21-111-11-1    linear of order 2
ρ34-2010-11    orthogonal faithful
ρ442010-1-1    orthogonal faithful
ρ5511-1-101    orthogonal faithful
ρ65-11-110-1    orthogonal faithful
ρ760-20010    orthogonal faithful

Permutation representations of S5
On 5 points: primitive, sharply 5-transitive - transitive group 5T5
Generators in S5
(1 2 3 4 5)
(1 5)

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

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

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

G:=TransitiveGroup(5,5);

On 6 points: primitive, sharply triply transitive - transitive group 6T14
Generators in S6
(2 3 4 5 6)
(1 3)(2 6)(4 5)

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

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

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

G:=TransitiveGroup(6,14);

On 10 points - transitive group 10T12
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 7)(3 6)(4 8)(5 9)

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

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

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

G:=TransitiveGroup(10,12);

On 10 points: primitive - transitive group 10T13
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(3 7)(5 8)(6 9)

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

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

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

G:=TransitiveGroup(10,13);

On 12 points - transitive group 12T74
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 10)(2 4)(3 8)(5 12)(6 11)(7 9)

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

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

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

G:=TransitiveGroup(12,74);

On 15 points - transitive group 15T10
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 13)(3 8)(5 14)(6 10)(7 15)(9 12)

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

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

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

G:=TransitiveGroup(15,10);

On 20 points - transitive group 20T30
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 12)(5 11)(6 15)(8 13)(9 18)(10 20)

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

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

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

G:=TransitiveGroup(20,30);

On 20 points - transitive group 20T32
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 11)(2 15)(3 17)(4 19)(5 12)(6 20)(7 16)(8 13)(9 18)(10 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), (1,11)(2,15)(3,17)(4,19)(5,12)(6,20)(7,16)(8,13)(9,18)(10,14)>;

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

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

G:=TransitiveGroup(20,32);

On 20 points - transitive group 20T35
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16)(2 17)(3 11)(5 12)(6 20)(7 9)(8 14)(10 18)(13 15)

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

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

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

G:=TransitiveGroup(20,35);

On 24 points - transitive group 24T202
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 17)(2 14)(3 5)(4 24)(6 11)(7 18)(8 16)(9 12)(10 21)(13 22)(15 23)(19 20)

G:=sub<Sym(24)| (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,17)(2,14)(3,5)(4,24)(6,11)(7,18)(8,16)(9,12)(10,21)(13,22)(15,23)(19,20)>;

G:=Group( (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,17)(2,14)(3,5)(4,24)(6,11)(7,18)(8,16)(9,12)(10,21)(13,22)(15,23)(19,20) );

G=PermutationGroup([[(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19),(20,21,22,23,24)], [(1,17),(2,14),(3,5),(4,24),(6,11),(7,18),(8,16),(9,12),(10,21),(13,22),(15,23),(19,20)]])

G:=TransitiveGroup(24,202);

On 30 points - transitive group 30T22
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)(2 16)(3 28)(4 18)(5 13)(6 7)(8 14)(9 17)(10 22)(11 29)(12 20)(15 21)(19 24)(25 27)(26 30)

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

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

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

G:=TransitiveGroup(30,22);

On 30 points - transitive group 30T25
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)(2 27)(3 26)(4 21)(5 13)(6 11)(7 15)(8 28)(9 22)(10 30)(12 18)(14 20)(16 25)(17 24)(19 29)

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

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

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

G:=TransitiveGroup(30,25);

On 30 points - transitive group 30T27
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 20)(3 11)(5 21)(6 16)(7 25)(9 27)(12 19)(13 30)(14 29)(15 22)(17 26)(24 28)

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

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

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

G:=TransitiveGroup(30,27);

S5 is a maximal subgroup of   ΓL2(𝔽4)
S5 is a maximal quotient of   CSU2(𝔽5)  C2.S5  A5⋊C4  ΓL2(𝔽4)

Polynomial with Galois group S5 over ℚ
actionf(x)Disc(f)
5T5x5-5x2+556·17
6T14x6-2x5+4x+2210·55
10T12x10-2x9-72x8-28x7+1334x6+1192x5-9936x4-8960x3+32052x2+18388x-39868220·52·114·972·1575·16012·99312
10T13x10-5x9-3x8+31x7-10x6-53x5+31x4+22x3-15x2-x+1432·613·3973
12T74x12-x10+2x8+4x6-3x4-3x2+1212·56·2934
15T10x15-5x14-5x13+65x12-55x11-301x10+550x9+505x8-1925x7+400x6+3450x5-3670x4-1690x3+6335x2-5195x+154726·522·78·136·172

Matrix representation of S5 in GL3(𝔽5) generated by

013
230
010
,
014
101
001
G:=sub<GL(3,GF(5))| [0,2,0,1,3,1,3,0,0],[0,1,0,1,0,0,4,1,1] >;

S5 in GAP, Magma, Sage, TeX

S_5
% in TeX

G:=Group("S5");
// GroupNames label

G:=SmallGroup(120,34);
// by ID

G=gap.SmallGroup(120,34);
# by ID

Export

Subgroup lattice of S5 in TeX
Character table of S5 in TeX

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