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G = C3⋊F5order 60 = 22·3·5

The semidirect product of C3 and F5 acting via F5/D5=C2

Aliases: C3⋊F5, C5⋊Dic3, C151C4, D5.S3, (C3×D5).1C2, SmallGroup(60,7)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3⋊F5
G = < a,b,c | a3=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C3⋊F5

 class 1 2 3 4A 4B 5 6 15A 15B size 1 5 2 15 15 4 10 4 4 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 -1 1 i -i 1 -1 1 1 linear of order 4 ρ4 1 -1 1 -i i 1 -1 1 1 linear of order 4 ρ5 2 2 -1 0 0 2 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 -1 0 0 2 1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ7 4 0 4 0 0 -1 0 -1 -1 orthogonal lifted from F5 ρ8 4 0 -2 0 0 -1 0 1-√-15/2 1+√-15/2 complex faithful ρ9 4 0 -2 0 0 -1 0 1+√-15/2 1-√-15/2 complex faithful

Permutation representations of C3⋊F5
On 15 points - transitive group 15T6
Generators in S15
```(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(2 3 5 4)(6 11)(7 13 10 14)(8 15 9 12)```

`G:=sub<Sym(15)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (2,3,5,4)(6,11)(7,13,10,14)(8,15,9,12)>;`

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

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

`G:=TransitiveGroup(15,6);`

On 30 points - transitive group 30T6
Generators in S30
```(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(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 17)(2 19 5 20)(3 16 4 18)(6 22)(7 24 10 25)(8 21 9 23)(11 27)(12 29 15 30)(13 26 14 28)```

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

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

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

`G:=TransitiveGroup(30,6);`

C3⋊F5 is a maximal subgroup of
S3×F5  C9⋊F5  C323F5  A4⋊F5  C75⋊C4  D5.D15  C15⋊F5  C152F5  ΓL2(𝔽4)  C5⋊Dic21
C3⋊F5 is a maximal quotient of
C15⋊C8  C9⋊F5  C323F5  A4⋊F5  C75⋊C4  D5.D15  C15⋊F5  C152F5  C5⋊Dic21

Polynomial with Galois group C3⋊F5 over ℚ
actionf(x)Disc(f)
15T6x15-30x10-3708x5-2214·320·530·231110

Matrix representation of C3⋊F5 in GL4(𝔽2) generated by

 0 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0
,
 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1
,
 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 1
`G:=sub<GL(4,GF(2))| [0,1,0,1,1,0,1,1,1,1,0,0,0,1,1,0],[0,0,0,1,1,0,0,1,1,1,0,1,1,1,1,1],[1,0,0,0,0,0,1,0,1,0,0,1,1,1,1,1] >;`

C3⋊F5 in GAP, Magma, Sage, TeX

`C_3\rtimes F_5`
`% in TeX`

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

`G:=SmallGroup(60,7);`
`// by ID`

`G=gap.SmallGroup(60,7);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-5,8,98,579,391]);`
`// Polycyclic`

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

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