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G = C22⋊S5order 480 = 25·3·5

The semidirect product of C22 and S5 acting via S5/A5=C2

non-abelian, not soluble

Aliases: C22⋊S5, A52D4, A5⋊C4⋊C2, (C2×S5)⋊2C2, C2.10(C2×S5), (C22×A5)⋊2C2, (C2×A5).4C22, SmallGroup(480,951)

Series: ChiefDerived Lower central Upper central

C1C2C22C22×A5 — C22⋊S5
A5C2×A5 — C22⋊S5
A5C2×A5 — C22⋊S5
C1C2C22

Subgroups: 1348 in 106 conjugacy classes, 9 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C5, S3 [×4], C6 [×3], C2×C4 [×3], D4 [×6], C23 [×6], D5 [×3], C10 [×2], Dic3, C12, A4, D6 [×7], C2×C6 [×2], C22⋊C4 [×3], C2×D4 [×3], C24, F5 [×2], D10 [×4], C2×C10, C4×S3, D12, C3⋊D4 [×2], C3×D4, S4, C2×A4 [×2], C22×S3 [×2], C22≀C2, C2×F5 [×2], C22×D5, A4⋊C4, S3×D4, C2×S4, C22×A4, A5, C22⋊F5, A4⋊D4, S5, C2×A5, C2×A5, A5⋊C4, C2×S5, C22×A5, C22⋊S5
Quotients: C1, C2 [×3], C22, D4, S5, C2×S5, C22⋊S5

Character table of C22⋊S5

 class 12A2B2C2D2E2F34A4B4C56A6B6C10A10B10C12
 size 11215152030202060602420404024242440
ρ11111111111111111111    trivial
ρ211-111-1-1111-111-1-1-1-111    linear of order 2
ρ311-1111-11-1-11111-1-1-11-1    linear of order 2
ρ411111-111-1-1-111-11111-1    linear of order 2
ρ52-20-220020002-20000-20    orthogonal lifted from D4
ρ644-400201-200-11-1-111-11    orthogonal lifted from C2×S5
ρ744400-201-200-1111-1-1-11    orthogonal lifted from S5
ρ844400201200-11-11-1-1-1-1    orthogonal lifted from S5
ρ944-400-201200-111-111-1-1    orthogonal lifted from C2×S5
ρ1055-511-1-1-11-110-1-110001    orthogonal lifted from C2×S5
ρ115551111-11-1-10-11-10001    orthogonal lifted from S5
ρ1255-5111-1-1-11-10-111000-1    orthogonal lifted from C2×S5
ρ1355511-11-1-1110-1-1-1000-1    orthogonal lifted from S5
ρ1466-6-2-20200001000-1-110    orthogonal lifted from C2×S5
ρ15666-2-20-2000010001110    orthogonal lifted from S5
ρ166-602-20000001000-55-10    orthogonal faithful
ρ176-602-200000010005-5-10    orthogonal faithful
ρ188-8000002000-2-2000020    orthogonal faithful
ρ1910-100-2200-200002000000    orthogonal faithful

Permutation representations of C22⋊S5
On 20 points - transitive group 20T116
Generators in S20
(1 13 5 15 4 19)(2 17)(3 11)(6 14 8 18 7 20)(9 12)(10 16)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,116);

On 20 points - transitive group 20T118
Generators in S20
(1 7 5)(2 16 6 18 10 12)(3 9)(4 14)(8 20)(11 17 13)(15 19)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,118);

On 24 points - transitive group 24T1341
Generators in S24
(1 18 9 20 7 22)(2 23 11 15 14 17)(3 13 21 10 19 12)(4 6 16 8 24 5)
(3 4)(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1341);

On 24 points - transitive group 24T1342
Generators in S24
(1 18 14 15 12 22)(2 23 6 20 9 17)(3 8 16 13 24 7)(4 11 21 5 19 10)
(3 4)(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1342);

On 24 points - transitive group 24T1349
Generators in S24
(1 9 22 21 20 13)(2 24 17 6 15 18)(3 14 7 16 5 8)(4 19 12 11 10 23)
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1349);

On 24 points - transitive group 24T1350
Generators in S24
(1 13 11 5 9 7)(2 20 6 22 14 24)(3 15 23 17 21 19)(4 8 18 10 16 12)
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1350);

Matrix representation of C22⋊S5 in GL6(ℤ)

-100000
010000
000010
0000-1-1
00-10-10
000-110
,
0-10000
-100000
000-100
000101
001100
000-110

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,1,-1,-1,1,0,0,0,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,1,1,-1,0,0,0,0,0,1,0,0,0,1,0,0] >;

C22⋊S5 in GAP, Magma, Sage, TeX

C_2^2\rtimes S_5
% in TeX

G:=Group("C2^2:S5");
// GroupNames label

G:=SmallGroup(480,951);
// by ID

G=gap.SmallGroup(480,951);
# by ID

Export

Character table of C22⋊S5 in TeX

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