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## G = C5⋊S4order 120 = 23·3·5

### The semidirect product of C5 and S4 acting via S4/A4=C2

Aliases: C5⋊S4, A4⋊D5, C22⋊D15, (C5×A4)⋊1C2, (C2×C10)⋊2S3, SmallGroup(120,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C5⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C5⋊S4
 Lower central C5×A4 — C5⋊S4
 Upper central C1

Generators and relations for C5⋊S4
G = < a,b,c,d,e | a5=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
30C2
4C3
15C22
15C4
20S3
3C10
6D5
4C15
15D4
3D10
3Dic5
4D15
5S4

Character table of C5⋊S4

 class 1 2A 2B 3 4 5A 5B 10A 10B 15A 15B 15C 15D size 1 3 30 8 30 2 2 6 6 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 2 0 -1 0 2 2 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ5 2 2 0 2 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 2 0 -1 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 orthogonal lifted from D15 ρ7 2 2 0 -1 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 orthogonal lifted from D15 ρ8 2 2 0 -1 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 orthogonal lifted from D15 ρ9 2 2 0 -1 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal lifted from D15 ρ10 3 -1 -1 0 1 3 3 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ11 3 -1 1 0 -1 3 3 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ12 6 -2 0 0 0 -3+3√5/2 -3-3√5/2 1-√5/2 1+√5/2 0 0 0 0 orthogonal faithful ρ13 6 -2 0 0 0 -3-3√5/2 -3+3√5/2 1+√5/2 1-√5/2 0 0 0 0 orthogonal faithful

Permutation representations of C5⋊S4
On 20 points - transitive group 20T33
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 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)
(2 5)(3 4)(6 11)(7 15)(8 14)(9 13)(10 12)(17 20)(18 19)```

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

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

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

`G:=TransitiveGroup(20,33);`

On 30 points - transitive group 30T19
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 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(2 5)(3 4)(6 23)(7 22)(8 21)(9 25)(10 24)(12 15)(13 14)(16 28)(17 27)(18 26)(19 30)(20 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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (2,5)(3,4)(6,23)(7,22)(8,21)(9,25)(10,24)(12,15)(13,14)(16,28)(17,27)(18,26)(19,30)(20,29)>;`

`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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (2,5)(3,4)(6,23)(7,22)(8,21)(9,25)(10,24)(12,15)(13,14)(16,28)(17,27)(18,26)(19,30)(20,29) );`

`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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(16,25),(17,21),(18,22),(19,23),(20,24)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,14,24),(7,15,25),(8,11,21),(9,12,22),(10,13,23)], [(2,5),(3,4),(6,23),(7,22),(8,21),(9,25),(10,24),(12,15),(13,14),(16,28),(17,27),(18,26),(19,30),(20,29)]])`

`G:=TransitiveGroup(30,19);`

On 30 points - transitive group 30T31
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 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)(21 27)(22 26)(23 30)(24 29)(25 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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,30)(24,29)(25,28)>;`

`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), (1,11)(2,12)(3,13)(4,14)(5,15)(16,25)(17,21)(18,22)(19,23)(20,24), (6,30)(7,26)(8,27)(9,28)(10,29)(16,25)(17,21)(18,22)(19,23)(20,24), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,14,24)(7,15,25)(8,11,21)(9,12,22)(10,13,23), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,30)(24,29)(25,28) );`

`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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(16,25),(17,21),(18,22),(19,23),(20,24)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,14,24),(7,15,25),(8,11,21),(9,12,22),(10,13,23)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20),(21,27),(22,26),(23,30),(24,29),(25,28)]])`

`G:=TransitiveGroup(30,31);`

C5⋊S4 is a maximal subgroup of   D5×S4  A4⋊D15  C42⋊D15  C244D15
C5⋊S4 is a maximal quotient of   Q8.D15  Q8⋊D15  A4⋊Dic5  C22⋊D45  A4⋊D15  C42⋊D15  C244D15

Matrix representation of C5⋊S4 in GL5(𝔽61)

 51 24 0 0 0 37 27 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 60 0 0 0 1 60 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 60 0 0 1 0 60 0 0 0 0 60
,
 60 60 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(61))| [51,37,0,0,0,24,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[60,1,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

C5⋊S4 in GAP, Magma, Sage, TeX

`C_5\rtimes S_4`
`% in TeX`

`G:=Group("C5:S4");`
`// GroupNames label`

`G:=SmallGroup(120,38);`
`// by ID`

`G=gap.SmallGroup(120,38);`
`# by ID`

`G:=PCGroup([5,-2,-3,-5,-2,2,41,362,1203,608,754,1134]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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