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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 15)(2 11)(3 18)(5 19)(6 9)(7 12)(8 14)(10 16)(17 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), (1,15)(2,11)(3,18)(5,19)(6,9)(7,12)(8,14)(10,16)(17,20)>;

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

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

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 15)(2 23)(3 10)(4 5)(6 12)(7 16)(8 19)(9 13)(11 20)(14 21)(17 24)(18 22)

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

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

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

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

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

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

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 15)(3 17)(5 29)(7 11)(8 28)(10 24)(12 23)(14 18)(16 30)(19 22)(20 21)(25 27)

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

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

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

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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