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G = C4×S5  order 480 = 25·3·5

Direct product of C4 and S5

direct product, non-abelian, not soluble

Aliases: C4×S5, CO3(𝔽5), (C2×S5).C2, A5⋊C4⋊2C2, (C4×A5)⋊3C2, A5⋊1(C2×C4), C2.1(C2×S5), (C2×A5).1C22, SmallGroup(480,943)

Series: Chief►Derived ►Lower central ►Upper central

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

Subgroups: 956 in 98 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, D10, C4×S3, C2×Dic3, C2×C12, S4, C2×A4, C22×S3, C4×D4, C4×D5, C2×F5, A4⋊C4, C4×A4, S3×C2×C4, C2×S4, A5, C4×F5, C4×S4, S5, C2×A5, A5⋊C4, C4×A5, C2×S5, C4×S5
Quotients: C1, C2, C4, C22, C2×C4, S5, C2×S5, C4×S5

Character table of C4×S5

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H4I4J56A6B6C1012A12B12C12D20A20B
 size 1110101515201110101515303030302420202024202020202424
ρ11111111111111111111111111111    trivial
ρ211-1-111111-1-111-1-1-1-11-11-1111-1-111    linear of order 2
ρ311-1-1111-1-111-1-111-1-11-11-11-1-111-1-1    linear of order 2
ρ41111111-1-1-1-1-1-1-1-11111111-1-1-1-1-1-1    linear of order 2
ρ51-11-1-111i-ii-ii-ii-i-1111-1-1-1-ii-iii-i    linear of order 4
ρ61-1-11-111i-i-iii-i-ii1-11-1-11-1-iii-ii-i    linear of order 4
ρ71-1-11-111-iii-i-iii-i1-11-1-11-1i-i-ii-ii    linear of order 4
ρ81-11-1-111-ii-ii-ii-ii-1111-1-1-1i-ii-i-ii    linear of order 4
ρ944220014422000000-1-11-1-111-1-1-1-1    orthogonal lifted from S5
ρ104422001-4-4-2-2000000-1-11-1-1-1-11111    orthogonal lifted from C2×S5
ρ1144-2-200144-2-2000000-1111-11111-1-1    orthogonal lifted from S5
ρ1244-2-2001-4-422000000-1111-1-1-1-1-111    orthogonal lifted from C2×S5
ρ134-42-20014i-4i2i-2i000000-1-1-111-iii-i-ii    complex faithful
ρ144-42-2001-4i4i-2i2i000000-1-1-111i-i-iii-i    complex faithful
ρ154-4-220014i-4i-2i2i000000-11-1-11-ii-ii-ii    complex faithful
ρ164-4-22001-4i4i2i-2i000000-11-1-11i-ii-ii-i    complex faithful
ρ1755-1-111-155-1-11111110-1-1-10-1-1-1-100    orthogonal lifted from S5
ρ18551111-1551111-1-1-1-101-110-1-11100    orthogonal lifted from S5
ρ1955-1-111-1-5-511-1-1-1-1110-1-1-10111100    orthogonal lifted from C2×S5
ρ20551111-1-5-5-1-1-1-111-1-101-11011-1-100    orthogonal lifted from C2×S5
ρ215-5-11-11-1-5i5ii-i-ii-ii-110-1110-ii-ii00    complex faithful
ρ225-51-1-11-1-5i5i-ii-iii-i1-1011-10-iii-i00    complex faithful
ρ235-51-1-11-15i-5ii-ii-i-ii1-1011-10i-i-ii00    complex faithful
ρ245-5-11-11-15i-5i-iii-ii-i-110-1110i-ii-i00    complex faithful
ρ256600-2-206600-2-2000010001000011    orthogonal lifted from S5
ρ266600-2-20-6-600220000100010000-1-1    orthogonal lifted from C2×S5
ρ276-6002-206i-6i00-2i2i00001000-10000i-i    complex faithful
ρ286-6002-20-6i6i002i-2i00001000-10000-ii    complex faithful

Permutation representations of C4×S5
►On 20 points - transitive group 20T123
Generators in S20
(1 8 12 15)(2 17 13 3)(4 9 18 6)(5 20 10 7)(11 16 14 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)| (1,8,12,15)(2,17,13,3)(4,9,18,6)(5,20,10,7)(11,16,14,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12,13,14)(15,16,17,18,19,20)>;
 
G:=Group( (1,8,12,15)(2,17,13,3)(4,9,18,6)(5,20,10,7)(11,16,14,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12,13,14)(15,16,17,18,19,20) );
 
G=PermutationGroup([[(1,8,12,15),(2,17,13,3),(4,9,18,6),(5,20,10,7),(11,16,14,19)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12,13,14),(15,16,17,18,19,20)]])
 
G:=TransitiveGroup(20,123);
 

►On 24 points - transitive group 24T1347
Generators in S24
(1 16 4 9)(2 17 23 14)(3 10 20 13)(5 12 22 15)(6 11 19 8)(7 21 18 24)

(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)| (1,16,4,9)(2,17,23,14)(3,10,20,13)(5,12,22,15)(6,11,19,8)(7,21,18,24), (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( (1,16,4,9)(2,17,23,14)(3,10,20,13)(5,12,22,15)(6,11,19,8)(7,21,18,24), (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([[(1,16,4,9),(2,17,23,14),(3,10,20,13),(5,12,22,15),(6,11,19,8),(7,21,18,24)], [(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(24,1347);
 

►On 24 points - transitive group 24T1348
Generators in S24
(1 15 13 9)(2 16 22 20)(4 24 8 6)(5 21)(7 11 17 23)(12 14)

(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)| (1,15,13,9)(2,16,22,20)(4,24,8,6)(5,21)(7,11,17,23)(12,14), (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( (1,15,13,9)(2,16,22,20)(4,24,8,6)(5,21)(7,11,17,23)(12,14), (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([[(1,15,13,9),(2,16,22,20),(4,24,8,6),(5,21),(7,11,17,23),(12,14)], [(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(24,1348);
 

Matrix representation of C4×S5 ►in GL3(𝔽5) generated by

301
340
300
,
033
032
334
G:=sub<GL(3,GF(5))| [3,3,3,0,4,0,1,0,0],[0,0,3,3,3,3,3,2,4] >;
 

C4×S5 in GAP, Magma, Sage, TeX 

C_4\times S_5
 
% in TeX
 
G:=Group("C4xS5");
 
// GroupNames label
 
G:=SmallGroup(480,943);
 
// by ID
 
G=gap.SmallGroup(480,943);
 
# by ID
 

Export

Character table of C4×S5 in TeX

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