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G = C33⋊C4order 108 = 22·33

2nd semidirect product of C33 and C4 acting faithfully

Aliases: C332C4, C323Dic3, C3⋊S3.S3, C3⋊(C32⋊C4), (C3×C3⋊S3).2C2, SmallGroup(108,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C33⋊C4
 Lower central C33 — C33⋊C4
 Upper central C1

Generators and relations for C33⋊C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C33⋊C4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 4A 4B 6 size 1 9 2 4 4 4 4 4 4 27 27 18 ρ1 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 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 i -i -1 linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 -i i -1 linear of order 4 ρ5 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 orthogonal lifted from S3 ρ6 2 -2 -1 -1 -1 2 2 -1 -1 0 0 1 symplectic lifted from Dic3, Schur index 2 ρ7 4 0 4 1 -2 1 -2 1 -2 0 0 0 orthogonal lifted from C32⋊C4 ρ8 4 0 4 -2 1 -2 1 -2 1 0 0 0 orthogonal lifted from C32⋊C4 ρ9 4 0 -2 1 -1+3√-3/2 -2 1 1 -1-3√-3/2 0 0 0 complex faithful ρ10 4 0 -2 -1+3√-3/2 1 1 -2 -1-3√-3/2 1 0 0 0 complex faithful ρ11 4 0 -2 1 -1-3√-3/2 -2 1 1 -1+3√-3/2 0 0 0 complex faithful ρ12 4 0 -2 -1-3√-3/2 1 1 -2 -1+3√-3/2 1 0 0 0 complex faithful

Permutation representations of C33⋊C4
On 12 points - transitive group 12T72
Generators in S12
```(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(2 10 5)(4 7 12)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)```

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

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

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

`G:=TransitiveGroup(12,72);`

On 18 points - transitive group 18T54
Generators in S18
```(1 17 15)(4 8 10)(6 14 12)
(1 17 15)(2 18 16)(3 9 7)(4 8 10)(5 11 13)(6 14 12)
(1 4 6)(2 5 3)(7 16 13)(8 14 17)(9 18 11)(10 12 15)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,54);`

On 27 points - transitive group 27T31
Generators in S27
```(1 6 4)(2 23 21)(3 15 13)(5 18 17)(7 19 16)(8 27 14)(9 26 20)(10 12 25)(11 22 24)
(1 19 17)(2 25 27)(3 9 11)(4 7 18)(5 6 16)(8 21 12)(10 14 23)(13 20 24)(15 26 22)
(1 3 2)(4 13 21)(5 22 14)(6 15 23)(7 20 12)(8 18 24)(9 25 19)(10 16 26)(11 27 17)
(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)```

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

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

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

`G:=TransitiveGroup(27,31);`

C33⋊C4 is a maximal subgroup of   S3×C32⋊C4  C33⋊D4  C33⋊Q8  C323Dic9  C34⋊C4  C62⋊Dic3
C33⋊C4 is a maximal quotient of   C334C8  C323Dic9  He34Dic3  C34⋊C4  C62⋊Dic3

Polynomial with Galois group C33⋊C4 over ℚ
actionf(x)Disc(f)
12T72x12+12x10-40x9+414x8-1416x7+3388x6-6552x5+8001x4-7448x3+7056x2+4704x+2744287·316·714·172·13612·15427272

Matrix representation of C33⋊C4 in GL4(𝔽7) generated by

 6 2 1 1 2 6 6 1 0 0 1 0 0 0 0 2
,
 6 2 5 1 0 4 0 5 4 4 0 6 0 0 0 2
,
 3 1 4 5 1 3 3 5 0 0 4 0 0 0 0 2
,
 1 3 1 4 3 6 4 5 1 6 3 2 5 5 1 4
`G:=sub<GL(4,GF(7))| [6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[1,3,1,5,3,6,6,5,1,4,3,1,4,5,2,4] >;`

C33⋊C4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes C_4`
`% in TeX`

`G:=Group("C3^3:C4");`
`// GroupNames label`

`G:=SmallGroup(108,37);`
`// by ID`

`G=gap.SmallGroup(108,37);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,3,-3,10,302,67,323,248,1804]);`
`// Polycyclic`

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

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