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G = C4xS5  order 480 = 25·3·5

Direct product of C4 and S5

direct product, non-abelian, not soluble

Aliases: C4xS5, CO3(F5), (C2xS5).C2, A5:C4:2C2, (C4xA5):3C2, A5:1(C2xC4), C2.1(C2xS5), (C2xA5).1C22, SmallGroup(480,943)

Series: Chief►Derived ►Lower central ►Upper central

Chief series
C1 — C2 — C4 — C4xA5 — C4xS5
Derived series
A5 — C4xS5
Lower central
A5 — C4xS5
Upper central
C1 — C4

Subgroups: 956 in 98 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2xC4, D4, C23, D5, C10, Dic3, C12, A4, D6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, F5, D10, C4xS3, C2xDic3, C2xC12, S4, C2xA4, C22xS3, C4xD4, C4xD5, C2xF5, A4:C4, C4xA4, S3xC2xC4, C2xS4, A5, C4xF5, C4xS4, S5, C2xA5, A5:C4, C4xA5, C2xS5, C4xS5
Quotients: C1, C2, C4, C22, C2xC4, S5, C2xS5, C4xS5

Character table of C4xS5

 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 C2xS5
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 C2xS5
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 C2xS5
ρ20551111-1-5-5-1-1-1-111-1-101-11011-1-100    orthogonal lifted from C2xS5
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 C2xS5
ρ276-6002-206i-6i00-2i2i00001000-10000i-i    complex faithful
ρ286-6002-20-6i6i002i-2i00001000-10000-ii    complex faithful

Permutation representations of C4xS5
►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 C4xS5 ►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] >;
 

C4xS5 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 C4xS5 in TeX

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