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## G = C34⋊C6order 486 = 2·35

### 1st semidirect product of C34 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C34⋊C6
 Chief series C1 — C3 — C32 — C33 — C34 — C32⋊He3 — C34⋊C6
 Lower central C32 — C33 — C34⋊C6
 Upper central C1 — C3 — C32

Generators and relations for C34⋊C6
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 832 in 153 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, C33, C33, C33, C32⋊C6, C2×He3, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C3×He3, C3×He3, C3×He3, C34, C3×C32⋊C6, S3×He3, C32×C3⋊S3, C32⋊He3, C34⋊C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C32⋊C6, C2×He3, S3×C32, C3×C32⋊C6, S3×He3, C34⋊C6

Permutation representations of C34⋊C6
On 18 points - transitive group 18T161
Generators in S18
```(1 9 17)(2 12 14)(3 10 16)(4 7 13)(5 8 18)(6 11 15)
(1 6 4)(2 5 3)(7 9 11)(8 10 12)(13 17 15)(14 18 16)
(1 9 17)(2 14 12)(3 16 10)(4 7 13)(5 18 8)(6 11 15)
(1 6 4)(2 3 5)(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,161);`

On 18 points - transitive group 18T164
Generators in S18
```(1 8 14)(2 15 9)(4 11 17)(5 18 12)
(1 8 14)(2 9 15)(3 10 16)(4 11 17)(5 12 18)(6 7 13)
(2 9 15)(3 10 16)(5 18 12)(6 13 7)
(1 14 8)(2 9 15)(3 16 10)(4 11 17)(5 18 12)(6 7 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,164);`

On 27 points - transitive group 27T180
Generators in S27
```(1 7 4)(2 5 8)(10 22 20)(11 21 23)(13 25 17)(14 18 26)
(1 7 4)(2 8 5)(3 9 6)(10 22 20)(11 23 21)(12 24 16)(13 25 17)(14 26 18)(15 27 19)
(1 22 25)(2 23 26)(4 10 13)(5 11 14)(7 20 17)(8 21 18)
(1 22 25)(2 26 23)(3 24 27)(4 10 13)(5 14 11)(6 12 15)(7 20 17)(8 18 21)(9 16 19)
(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)```

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

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

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

`G:=TransitiveGroup(27,180);`

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H ··· 3R 3S ··· 3X 3Y ··· 3AD 6A 6B 6C ··· 6J order 1 2 3 3 3 3 3 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 size 1 9 1 1 2 2 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 9 9 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 He3 C2×He3 C32⋊C6 C3×C32⋊C6 S3×He3 C34⋊C6 kernel C34⋊C6 C32⋊He3 C3×C32⋊C6 C32×C3⋊S3 C3×He3 C34 C3×He3 C33 C3⋊S3 C32 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 8 2 2 1 2 2 6

Matrix representation of C34⋊C6 in GL6(𝔽7)

 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 4 1 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0

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

C34⋊C6 in GAP, Magma, Sage, TeX

`C_3^4\rtimes C_6`
`% in TeX`

`G:=Group("C3^4:C6");`
`// GroupNames label`

`G:=SmallGroup(486,102);`
`// by ID`

`G=gap.SmallGroup(486,102);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,3244,3250,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;`
`// generators/relations`

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