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## G = C3⋊F5⋊S3order 360 = 23·32·5

### The semidirect product of C3⋊F5 and S3 acting via S3/C3=C2

Aliases: C3⋊F5⋊S3, D5.2S32, C3⋊S31F5, C32(S3×F5), C152(C4×S3), C3⋊D153C4, C5⋊(C6.D6), C323(C2×F5), (C3×D5).3D6, (C32×D5).4C22, (C3×C3⋊F5)⋊3C2, (C5×C3⋊S3)⋊3C4, (C3×C15)⋊6(C2×C4), (D5×C3⋊S3).4C2, SmallGroup(360,129)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C3⋊F5⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C3×C3⋊F5 — C3⋊F5⋊S3
 Lower central C3×C15 — C3⋊F5⋊S3
 Upper central C1

Generators and relations for C3⋊F5⋊S3
G = < a,b,c,d,e | a3=b5=c4=d3=e2=1, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=b3, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 548 in 74 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C3, C3, C4, C22, C5, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C15, C15, C3⋊S3, C3⋊S3, C3×C6, F5, D10, C4×S3, C5×S3, C3×D5, C3×D5, D15, C3×Dic3, C2×C3⋊S3, C2×F5, C3×C15, C3×F5, C3⋊F5, S3×D5, C6.D6, C32×D5, C5×C3⋊S3, C3⋊D15, S3×F5, C3×C3⋊F5, D5×C3⋊S3, C3⋊F5⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, F5, C4×S3, S32, C2×F5, C6.D6, S3×F5, C3⋊F5⋊S3

Character table of C3⋊F5⋊S3

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5 6A 6B 6C 10 12A 12B 12C 12D 15A 15B 15C 15D size 1 5 9 45 2 2 4 15 15 15 15 4 10 10 20 36 30 30 30 30 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 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 1 1 -1 -1 1 -1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 -1 1 1 1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 i i -i -i 1 -1 -1 -1 -1 i -i -i i 1 1 1 1 linear of order 4 ρ6 1 -1 1 -1 1 1 1 -i i -i i 1 -1 -1 -1 1 -i -i i i 1 1 1 1 linear of order 4 ρ7 1 -1 -1 1 1 1 1 -i -i i i 1 -1 -1 -1 -1 -i i i -i 1 1 1 1 linear of order 4 ρ8 1 -1 1 -1 1 1 1 i -i i -i 1 -1 -1 -1 1 i i -i -i 1 1 1 1 linear of order 4 ρ9 2 2 0 0 2 -1 -1 -2 0 0 -2 2 2 -1 -1 0 1 0 1 0 -1 -1 -1 2 orthogonal lifted from D6 ρ10 2 2 0 0 -1 2 -1 0 -2 -2 0 2 -1 2 -1 0 0 1 0 1 2 -1 -1 -1 orthogonal lifted from D6 ρ11 2 2 0 0 -1 2 -1 0 2 2 0 2 -1 2 -1 0 0 -1 0 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 0 0 2 -1 -1 2 0 0 2 2 2 -1 -1 0 -1 0 -1 0 -1 -1 -1 2 orthogonal lifted from S3 ρ13 2 -2 0 0 -1 2 -1 0 2i -2i 0 2 1 -2 1 0 0 i 0 -i 2 -1 -1 -1 complex lifted from C4×S3 ρ14 2 -2 0 0 -1 2 -1 0 -2i 2i 0 2 1 -2 1 0 0 -i 0 i 2 -1 -1 -1 complex lifted from C4×S3 ρ15 2 -2 0 0 2 -1 -1 -2i 0 0 2i 2 -2 1 1 0 i 0 -i 0 -1 -1 -1 2 complex lifted from C4×S3 ρ16 2 -2 0 0 2 -1 -1 2i 0 0 -2i 2 -2 1 1 0 -i 0 i 0 -1 -1 -1 2 complex lifted from C4×S3 ρ17 4 0 4 0 4 4 4 0 0 0 0 -1 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ18 4 0 -4 0 4 4 4 0 0 0 0 -1 0 0 0 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from C2×F5 ρ19 4 4 0 0 -2 -2 1 0 0 0 0 4 -2 -2 1 0 0 0 0 0 -2 1 1 -2 orthogonal lifted from S32 ρ20 4 -4 0 0 -2 -2 1 0 0 0 0 4 2 2 -1 0 0 0 0 0 -2 1 1 -2 orthogonal lifted from C6.D6 ρ21 8 0 0 0 -4 8 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 1 1 1 orthogonal lifted from S3×F5 ρ22 8 0 0 0 8 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 1 1 1 -2 orthogonal lifted from S3×F5 ρ23 8 0 0 0 -4 -4 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 1 -1+3√5/2 -1-3√5/2 1 orthogonal faithful ρ24 8 0 0 0 -4 -4 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 1 -1-3√5/2 -1+3√5/2 1 orthogonal faithful

Permutation representations of C3⋊F5⋊S3
On 30 points - transitive group 30T86
Generators in S30
```(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 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 18)(2 20 5 16)(3 17 4 19)(6 22 7 24)(8 21 10 25)(9 23)(11 27 12 29)(13 26 15 30)(14 28)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)```

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

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

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

`G:=TransitiveGroup(30,86);`

Matrix representation of C3⋊F5⋊S3 in GL8(𝔽61)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 1 0 0 0 0 0 60 0 0 1 0 0 0 0 60 0 0 0
,
 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 0 10 51 54 0 0 0 0 0 3 51 0 10 0 0 0 0 10 0 51 3 0 0 0 0 0 54 51 10
,
 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60

`G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,10,3,10,0,0,0,0,0,51,51,0,54,0,0,0,0,54,0,51,51,0,0,0,0,0,10,3,10],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;`

C3⋊F5⋊S3 in GAP, Magma, Sage, TeX

`C_3\rtimes F_5\rtimes S_3`
`% in TeX`

`G:=Group("C3:F5:S3");`
`// GroupNames label`

`G:=SmallGroup(360,129);`
`// by ID`

`G=gap.SmallGroup(360,129);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,201,730,7781,2609]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^5=c^4=d^3=e^2=1,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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