S5 - GroupNames

G = S₅ order 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: Chief►Derived ►Lower central ►Upper central

C₁ — A₅ — S₅

A₅ — S₅

A₅ — S₅

C₁

₁₀C₂

₁₅C₂

₁₀C₃

₆C₅

₅C₂²

₁₅C₂²

₁₅C₄

₁₀C₆

₁₀S₃

₆D₅

₁₅D₄

₅A₄

₁₀D₆

₆F₅

₅S₄

A₅

S₅

Character table of S₅

class	1	2A	2B	3	4	5	6
size	1	10	15	20	30	24	20
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	1	-1	linear of order 2
ρ₃	4	-2	0	1	0	-1	1	orthogonal faithful
ρ₄	4	2	0	1	0	-1	-1	orthogonal faithful
ρ₅	5	1	1	-1	-1	0	1	orthogonal faithful
ρ₆	5	-1	1	-1	1	0	-1	orthogonal faithful
ρ₇	6	0	-2	0	0	1	0	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

action	f(x)	Disc(f)
5T5	x⁵-5x²+5	5⁶·17
6T14	x⁶-2x⁵+4x+2	2¹⁰·5⁵
10T12	x¹⁰-2x⁹-72x⁸-28x⁷+1334x⁶+1192x⁵-9936x⁴-8960x³+32052x²+18388x-39868	2²⁰·5²·11⁴·97²·157⁵·1601²·9931²
10T13	x¹⁰-5x⁹-3x⁸+31x⁷-10x⁶-53x⁵+31x⁴+22x³-15x²-x+1	43²·61³·397³
12T74	x¹²-x¹⁰+2x⁸+4x⁶-3x⁴-3x²+1	2¹²·5⁶·293⁴
15T10	x¹⁵-5x¹⁴-5x¹³+65x¹²-55x¹¹-301x¹⁰+550x⁹+505x⁸-1925x⁷+400x⁶+3450x⁵-3670x⁴-1690x³+6335x²-5195x+1547	2⁶·5²²·7⁸·13⁶·17²

Matrix representation of S₅ ►in GL₃(𝔽₅) generated by

0	1	3
2	3	0
0	1	0

0	1	4
1	0	1
0	0	1

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

G = S5 order 120 = 23·3·5

Symmetric group on 5 letters

G = S₅ order 120 = 2³·3·5