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## G = C32⋊C4order 36 = 22·32

### The semidirect product of C32 and C4 acting faithfully

Aliases: C32⋊C4, C3⋊S3.C2, SmallGroup(36,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4
 Lower central C32 — C32⋊C4
 Upper central C1

Generators and relations for C32⋊C4
G = < a,b,c | a3=b3=c4=1, cbc-1=ab=ba, cac-1=a-1b >

Character table of C32⋊C4

 class 1 2 3A 3B 4A 4B size 1 9 4 4 9 9 ρ1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 linear of order 2 ρ3 1 -1 1 1 i -i linear of order 4 ρ4 1 -1 1 1 -i i linear of order 4 ρ5 4 0 1 -2 0 0 orthogonal faithful ρ6 4 0 -2 1 0 0 orthogonal faithful

Permutation representations of C32⋊C4
On 6 points - transitive group 6T10
Generators in S6
```(1 6 4)(2 3 5)
(1 4 6)
(1 2)(3 4 5 6)```

`G:=sub<Sym(6)| (1,6,4)(2,3,5), (1,4,6), (1,2)(3,4,5,6)>;`

`G:=Group( (1,6,4)(2,3,5), (1,4,6), (1,2)(3,4,5,6) );`

`G=PermutationGroup([[(1,6,4),(2,3,5)], [(1,4,6)], [(1,2),(3,4,5,6)]])`

`G:=TransitiveGroup(6,10);`

On 9 points: primitive - transitive group 9T9
Generators in S9
```(1 8 6)(2 9 5)(3 7 4)
(1 4 2)(3 9 8)(5 6 7)
(2 3 4 5)(6 7 8 9)```

`G:=sub<Sym(9)| (1,8,6)(2,9,5)(3,7,4), (1,4,2)(3,9,8)(5,6,7), (2,3,4,5)(6,7,8,9)>;`

`G:=Group( (1,8,6)(2,9,5)(3,7,4), (1,4,2)(3,9,8)(5,6,7), (2,3,4,5)(6,7,8,9) );`

`G=PermutationGroup([[(1,8,6),(2,9,5),(3,7,4)], [(1,4,2),(3,9,8),(5,6,7)], [(2,3,4,5),(6,7,8,9)]])`

`G:=TransitiveGroup(9,9);`

On 12 points - transitive group 12T17
Generators in S12
```(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(2 10 5)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)```

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

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

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

`G:=TransitiveGroup(12,17);`

On 18 points - transitive group 18T10
Generators in S18
```(1 14 12)(2 8 10)(3 16 7)(4 15 11)(5 9 18)(6 13 17)
(1 3 5)(2 17 15)(4 10 13)(6 11 8)(7 18 12)(9 14 16)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)```

`G:=sub<Sym(18)| (1,14,12)(2,8,10)(3,16,7)(4,15,11)(5,9,18)(6,13,17), (1,3,5)(2,17,15)(4,10,13)(6,11,8)(7,18,12)(9,14,16), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;`

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

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

`G:=TransitiveGroup(18,10);`

C32⋊C4 is a maximal subgroup of
F9  S3≀C2  PSU3(𝔽2)  C32⋊F5  C92⋊C4  C344C4  A6  (C3×C39)⋊C4
C32⋊Dicp: C33⋊C4  C32⋊Dic5  C32⋊Dic7  C32⋊Dic11  C32⋊Dic13 ...
C32⋊C4 is a maximal quotient of
C322C8  He3⋊C4  C32⋊F5  C92⋊C4  C344C4  (C3×C39)⋊C4
C32⋊Dicp: C33⋊C4  C32⋊Dic5  C32⋊Dic7  C32⋊Dic11  C32⋊Dic13 ...

Polynomial with Galois group C32⋊C4 over ℚ
actionf(x)Disc(f)
6T10x6-21x4+21x3+99x2-198x+99312·54·112
9T9x9-36x7-45x6+297x5+459x4-858x3-1404x2+819x+1339326·136·292·432
12T17x12-48x10-72x9+636x8+1848x7-484x6-6528x5-5625x4+3456x3+6744x2+2880x+376253·314·74·172·472·2392·7875292

Matrix representation of C32⋊C4 in GL4(ℤ) generated by

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

C32⋊C4 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_4`
`% in TeX`

`G:=Group("C3^2:C4");`
`// GroupNames label`

`G:=SmallGroup(36,9);`
`// by ID`

`G=gap.SmallGroup(36,9);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,3,8,338,54,515,199]);`
`// Polycyclic`

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

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