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G = C4⋊S5  order 480 = 25·3·5

The semidirect product of C4 and S5 acting via S5/A5=C2

non-abelian, not soluble

Aliases: C4⋊S5, A5⋊1D4, (C2×S5)⋊1C2, (C4×A5)⋊2C2, C2.3(C2×S5), (C2×A5).2C22, SmallGroup(480,944)

Series: Chief►Derived ►Lower central ►Upper central

Chief series
C1 — C2 — C4 — C4×A5 — C4⋊S5
Derived series
A5 — C2×A5 — C4⋊S5
Lower central
A5 — C2×A5 — C4⋊S5
Upper central
C1 — C2 — C4

Subgroups: 1188 in 96 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, D10, C4×S3, D12, C3⋊D4, C3×D4, S4, C2×A4, C22×S3, C4⋊D4, C4×D5, C2×F5, C4×A4, S3×D4, C2×S4, A5, C4⋊F5, C4⋊S4, S5, C2×A5, C4×A5, C2×S5, C4⋊S5
Quotients: C1, C2, C22, D4, S5, C2×S5, C4⋊S5

Character table of C4⋊S5

 class 12A2B2C2D2E34A4B4C4D56A6B6C101220A20B
 size 11151520202023060602420404024402424
ρ11111111111111111111    trivial
ρ21111-1-1111-1-111-1-11111    linear of order 2
ρ311111-11-1-1-11111-11-1-1-1    linear of order 2
ρ41111-111-1-11-111-111-1-1-1    linear of order 2
ρ52-22-200200002-200-2000    orthogonal lifted from D4
ρ64400-2-214000-1111-11-1-1    orthogonal lifted from S5
ρ744002-21-4000-11-11-1-111    orthogonal lifted from C2×S5
ρ844002214000-11-1-1-11-1-1    orthogonal lifted from S5
ρ94400-221-4000-111-1-1-111    orthogonal lifted from C2×S5
ρ10551111-151-1-10-1110-100    orthogonal lifted from S5
ρ1155111-1-1-5-11-10-11-10100    orthogonal lifted from C2×S5
ρ125511-11-1-5-1-110-1-110100    orthogonal lifted from C2×S5
ρ135511-1-1-151110-1-1-10-100    orthogonal lifted from S5
ρ1466-2-20006-20010001011    orthogonal lifted from S5
ρ1566-2-2000-6200100010-1-1    orthogonal lifted from C2×S5
ρ166-6-2200000001000-10-√-5√-5    complex faithful
ρ176-6-2200000001000-10√-5-√-5    complex faithful
ρ188-8000020000-2-2002000    orthogonal faithful
ρ1910-102-200-2000002000000    orthogonal faithful

Permutation representations of C4⋊S5
►On 20 points - transitive group 20T120
Generators in S20
(1 16)(2 12)(3 20)(4 8)(5 9)(7 11)(14 18)

(1 2)(3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)
 
G:=sub<Sym(20)| (1,16)(2,12)(3,20)(4,8)(5,9)(7,11)(14,18), (1,2)(3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)>;
 
G:=Group( (1,16)(2,12)(3,20)(4,8)(5,9)(7,11)(14,18), (1,2)(3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20) );
 
G=PermutationGroup([[(1,16),(2,12),(3,20),(4,8),(5,9),(7,11),(14,18)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20)]])
 
G:=TransitiveGroup(20,120);
 

►On 24 points - transitive group 24T1352
Generators in S24
(1 19)(2 23)(3 13)(4 12)(5 9)(6 14)(7 24)(8 20)(10 18)(11 15)(16 21)(17 22)

(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)| (1,19)(2,23)(3,13)(4,12)(5,9)(6,14)(7,24)(8,20)(10,18)(11,15)(16,21)(17,22), (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( (1,19)(2,23)(3,13)(4,12)(5,9)(6,14)(7,24)(8,20)(10,18)(11,15)(16,21)(17,22), (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([[(1,19),(2,23),(3,13),(4,12),(5,9),(6,14),(7,24),(8,20),(10,18),(11,15),(16,21),(17,22)], [(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(24,1352);
 

Matrix representation of C4⋊S5 ►in GL6(𝔽3)

022210
000001
011002
202021
010021
010000
,
020012
212000
202021
022210
001101
120101

G:=sub<GL(6,GF(3))| [0,0,0,2,0,0,2,0,1,0,1,1,2,0,1,2,0,0,2,0,0,0,0,0,1,0,0,2,2,0,0,1,2,1,1,0],[0,2,2,0,0,1,2,1,0,2,0,2,0,2,2,2,1,0,0,0,0,2,1,1,1,0,2,1,0,0,2,0,1,0,1,1] >;
 

C4⋊S5 in GAP, Magma, Sage, TeX 

C_4\rtimes S_5
 
% in TeX
 
G:=Group("C4:S5");
 
// GroupNames label
 
G:=SmallGroup(480,944);
 
// by ID
 
G=gap.SmallGroup(480,944);
 
# by ID
 

Export

Character table of C4⋊S5 in TeX

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