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## G = C32⋊D20order 360 = 23·32·5

### The semidirect product of C32 and D20 acting via D20/C5=D4

Aliases: C32⋊D20, C51S3≀C2, C32⋊C4⋊D5, (C3×C15)⋊2D4, D15⋊S32C2, C3⋊S3.2D10, (C5×C32⋊C4)⋊1C2, (C5×C3⋊S3).3C22, SmallGroup(360,134)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3⋊S3 — C32⋊D20
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — D15⋊S3 — C32⋊D20
 Lower central C3×C15 — C5×C3⋊S3 — C32⋊D20
 Upper central C1

Generators and relations for C32⋊D20
G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

9C2
30C2
30C2
2C3
2C3
9C4
45C22
45C22
6S3
6S3
10S3
10S3
30C6
30C6
6D5
6D5
9C10
2C15
2C15
45D4
30D6
30D6
10C3×S3
10C3×S3
9D10
9D10
9C20
2D15
2D15
5S32
5S32
9D20

Character table of C32⋊D20

 class 1 2A 2B 2C 3A 3B 4 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D size 1 9 30 30 4 4 18 2 2 60 60 18 18 8 8 8 8 18 18 18 18 ρ1 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 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 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 linear of order 2 ρ5 2 -2 0 0 2 2 0 2 2 0 0 -2 -2 2 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 2 2 -2 -1-√5/2 -1+√5/2 0 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 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ7 2 2 0 0 2 2 2 -1-√5/2 -1+√5/2 0 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 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 0 0 2 2 2 -1+√5/2 -1-√5/2 0 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 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 2 0 0 2 2 -2 -1+√5/2 -1-√5/2 0 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 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ10 2 -2 0 0 2 2 0 -1+√5/2 -1-√5/2 0 0 1-√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ11 2 -2 0 0 2 2 0 -1-√5/2 -1+√5/2 0 0 1+√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ12 2 -2 0 0 2 2 0 -1-√5/2 -1+√5/2 0 0 1+√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ13 2 -2 0 0 2 2 0 -1+√5/2 -1-√5/2 0 0 1-√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ14 4 0 2 0 -2 1 0 4 4 -1 0 0 0 1 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 0 0 -2 1 -2 0 4 4 0 1 0 0 -2 1 1 -2 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 0 -2 0 -2 1 0 4 4 1 0 0 0 1 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 0 0 2 1 -2 0 4 4 0 -1 0 0 -2 1 1 -2 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 8 0 0 0 -4 2 0 -2-2√5 -2+2√5 0 0 0 0 -1+√5/2 1-√5 1+√5 -1-√5/2 0 0 0 0 orthogonal faithful ρ19 8 0 0 0 2 -4 0 -2-2√5 -2+2√5 0 0 0 0 1-√5 -1+√5/2 -1-√5/2 1+√5 0 0 0 0 orthogonal faithful ρ20 8 0 0 0 -4 2 0 -2+2√5 -2-2√5 0 0 0 0 -1-√5/2 1+√5 1-√5 -1+√5/2 0 0 0 0 orthogonal faithful ρ21 8 0 0 0 2 -4 0 -2+2√5 -2-2√5 0 0 0 0 1+√5 -1-√5/2 -1+√5/2 1-√5 0 0 0 0 orthogonal faithful

Permutation representations of C32⋊D20
On 30 points - transitive group 30T95
Generators in S30
```(2 27 17)(4 19 29)(6 11 21)(8 23 13)(10 15 25)
(1 26 16)(3 18 28)(5 30 20)(7 22 12)(9 14 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 25 26 27 28 29 30)
(1 9)(2 8)(3 7)(4 6)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)```

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

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

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

`G:=TransitiveGroup(30,95);`

Matrix representation of C32⋊D20 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 1 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 15 9 0 0 0 0 52 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0
,
 9 46 0 0 0 0 46 52 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 60 0 0 0 0 0 0 60 0 0

`G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[15,52,0,0,0,0,9,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[9,46,0,0,0,0,46,52,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,60,0,0,0,0,0,0,60,0,0] >;`

C32⋊D20 in GAP, Magma, Sage, TeX

`C_3^2\rtimes D_{20}`
`% in TeX`

`G:=Group("C3^2:D20");`
`// GroupNames label`

`G:=SmallGroup(360,134);`
`// by ID`

`G=gap.SmallGroup(360,134);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-5,73,31,579,585,111,244,130,376,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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