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## G = C32⋊F5⋊C2order 360 = 23·32·5

### The semidirect product of C32⋊F5 and C2 acting faithfully

Aliases: C32⋊F5⋊C2, C3⋊S32F5, D5⋊(C32⋊C4), C321(C2×F5), C3⋊D15.C22, (C32×D5)⋊2C4, C51(C2×C32⋊C4), (C5×C3⋊S3)⋊1C4, (C3×C15)⋊2(C2×C4), (D5×C3⋊S3).2C2, SmallGroup(360,131)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C32⋊F5⋊C2
 Chief series C1 — C5 — C3×C15 — C3⋊D15 — C32⋊F5 — C32⋊F5⋊C2
 Lower central C3×C15 — C32⋊F5⋊C2
 Upper central C1

Generators and relations for C32⋊F5⋊C2
G = < a,b,c,d,e | a3=b3=c5=d4=e2=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, eae=a-1, bc=cb, ebe=b-1, dcd-1=c3, ce=ec, de=ed >

Character table of C32⋊F5⋊C2

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5 6A 6B 10 15A 15B 15C 15D size 1 5 9 45 4 4 45 45 45 45 4 20 20 36 8 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 1 i -i i -i 1 -1 -1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -i -i i i 1 1 1 -1 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 1 i i -i -i 1 1 1 -1 1 1 1 1 linear of order 4 ρ8 1 -1 1 -1 1 1 -i i -i i 1 -1 -1 1 1 1 1 1 linear of order 4 ρ9 4 -4 0 0 1 -2 0 0 0 0 4 2 -1 0 -2 1 -2 1 orthogonal lifted from C2×C32⋊C4 ρ10 4 -4 0 0 -2 1 0 0 0 0 4 -1 2 0 1 -2 1 -2 orthogonal lifted from C2×C32⋊C4 ρ11 4 0 4 0 4 4 0 0 0 0 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ12 4 4 0 0 -2 1 0 0 0 0 4 1 -2 0 1 -2 1 -2 orthogonal lifted from C32⋊C4 ρ13 4 4 0 0 1 -2 0 0 0 0 4 -2 1 0 -2 1 -2 1 orthogonal lifted from C32⋊C4 ρ14 4 0 -4 0 4 4 0 0 0 0 -1 0 0 1 -1 -1 -1 -1 orthogonal lifted from C2×F5 ρ15 8 0 0 0 2 -4 0 0 0 0 -2 0 0 0 1 -1-3√5/2 1 -1+3√5/2 orthogonal faithful ρ16 8 0 0 0 2 -4 0 0 0 0 -2 0 0 0 1 -1+3√5/2 1 -1-3√5/2 orthogonal faithful ρ17 8 0 0 0 -4 2 0 0 0 0 -2 0 0 0 -1+3√5/2 1 -1-3√5/2 1 orthogonal faithful ρ18 8 0 0 0 -4 2 0 0 0 0 -2 0 0 0 -1-3√5/2 1 -1+3√5/2 1 orthogonal faithful

Permutation representations of C32⋊F5⋊C2
On 30 points - transitive group 30T97
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 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)
(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 27 12 24)(7 29 11 22)(8 26 15 25)(9 28 14 23)(10 30 13 21)
(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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (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)(6,27,12,24)(7,29,11,22)(8,26,15,25)(9,28,14,23)(10,30,13,21), (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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (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)(6,27,12,24)(7,29,11,22)(8,26,15,25)(9,28,14,23)(10,30,13,21), (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)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13)], [(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),(6,27,12,24),(7,29,11,22),(8,26,15,25),(9,28,14,23),(10,30,13,21)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30)]])`

`G:=TransitiveGroup(30,97);`

Matrix representation of C32⋊F5⋊C2 in GL8(𝔽61)

 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 0 0 0 0 0 0 0 1 5 0 0 0 0 1 1 36 59
,
 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 0 60 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 56 1 5 0 0 0 0 1 1 36 59
,
 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 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
,
 60 1 60 0 0 0 0 0 59 1 0 60 0 0 0 0 60 0 1 59 0 0 0 0 0 60 1 60 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 36 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 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 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 60 60 25 1

`G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,36,0,0,0,0,0,0,5,59],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,60,60,56,1,0,0,0,0,0,0,1,36,0,0,0,0,0,0,5,59],[60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,59,60,0,0,0,0,0,1,1,0,60,0,0,0,0,60,0,1,1,0,0,0,0,0,60,59,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,36,60,0,0,0,0,0,1,60,0,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,60,0,0,0,0,60,0,0,60,0,0,0,0,0,0,60,25,0,0,0,0,0,0,0,1] >;`

C32⋊F5⋊C2 in GAP, Magma, Sage, TeX

`C_3^2\rtimes F_5\rtimes C_2`
`% in TeX`

`G:=Group("C3^2:F5:C2");`
`// GroupNames label`

`G:=SmallGroup(360,131);`
`// by ID`

`G=gap.SmallGroup(360,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,963,201,111,964,730,376,7781,2609]);`
`// Polycyclic`

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

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