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## G = C34.C6order 486 = 2·35

### 4th non-split extension by C34 of C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C34.C6
 Chief series C1 — C3 — C32 — C33 — C34 — C34.C3 — 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=1, e6=b, 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: 472 in 108 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3 [×2], C3 [×8], S3 [×2], C6 [×2], C9 [×6], C32 [×3], C32 [×27], C18 [×3], C3×S3 [×6], C3⋊S3, C3×C6, C3×C9 [×3], C3×C9 [×3], 3- 1+2 [×5], C33 [×2], C33 [×8], S3×C9 [×3], C2×3- 1+2, S3×C32 [×2], C3×C3⋊S3, C3×C3⋊S3, C32⋊C9 [×3], C32⋊C9 [×3], C3×3- 1+2, C3×3- 1+2, C34, C32⋊C18 [×3], S3×3- 1+2, C32×C3⋊S3, C34.C3, C34.C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, 3- 1+2, C32⋊C6, C2×3- 1+2, S3×C32, C3×C32⋊C6, S3×3- 1+2, C34.C6

Permutation representations of C34.C6
On 18 points - transitive group 18T160
Generators in S18
```(2 8 14)(3 15 9)(5 11 17)(6 18 12)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)
(2 8 14)(3 9 15)(5 17 11)(6 18 12)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,160);`

On 27 points - transitive group 27T208
Generators in S27
```(2 8 5)(3 6 9)(10 16 22)(11 23 17)(13 19 25)(14 26 20)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(2 10 19)(3 11 20)(5 22 13)(6 23 14)(8 16 25)(9 17 26)
(1 18 27)(2 10 19)(3 20 11)(4 12 21)(5 22 13)(6 14 23)(7 24 15)(8 16 25)(9 26 17)
(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)| (2,8,5)(3,6,9)(10,16,22)(11,23,17)(13,19,25)(14,26,20), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (2,10,19)(3,11,20)(5,22,13)(6,23,14)(8,16,25)(9,17,26), (1,18,27)(2,10,19)(3,20,11)(4,12,21)(5,22,13)(6,14,23)(7,24,15)(8,16,25)(9,26,17), (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( (2,8,5)(3,6,9)(10,16,22)(11,23,17)(13,19,25)(14,26,20), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (2,10,19)(3,11,20)(5,22,13)(6,23,14)(8,16,25)(9,17,26), (1,18,27)(2,10,19)(3,20,11)(4,12,21)(5,22,13)(6,14,23)(7,24,15)(8,16,25)(9,26,17), (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([(2,8,5),(3,6,9),(10,16,22),(11,23,17),(13,19,25),(14,26,20)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(2,10,19),(3,11,20),(5,22,13),(6,23,14),(8,16,25),(9,17,26)], [(1,18,27),(2,10,19),(3,20,11),(4,12,21),(5,22,13),(6,14,23),(7,24,15),(8,16,25),(9,26,17)], [(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(27,208);`

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H ··· 3R 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 ··· 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 3 3 6 ··· 6 9 9 27 27 9 ··· 9 18 ··· 18 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 C32⋊C6 C3×C32⋊C6 S3×3- 1+2 C34.C6 kernel C34.C6 C34.C3 C32⋊C18 C32×C3⋊S3 C32⋊C9 C34 C3×3- 1+2 C3×C9 C33 C3⋊S3 C32 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 2 2 1 2 2 6

Matrix representation of C34.C6 in GL6(𝔽19)

 11 0 0 0 0 0 0 1 0 0 0 0 18 1 7 0 0 0 0 6 0 11 0 0 4 0 0 0 1 0 14 14 0 0 0 7
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 11 0 0 0 0 11 12 7 0 0 0 0 13 0 1 0 0 9 17 0 0 7 0 3 0 0 0 0 11
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 2 9 0 7 0 0 13 17 0 0 7 0 14 16 0 0 0 7
,
 14 16 0 0 0 15 14 6 0 10 0 0 2 5 0 11 1 1 8 17 11 13 0 17 12 11 0 14 0 6 10 6 0 2 0 5

`G:=sub<GL(6,GF(19))| [11,0,18,0,4,14,0,1,1,6,0,14,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,11,0,9,3,0,11,12,13,17,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[11,0,0,2,13,14,0,11,0,9,17,16,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[14,14,2,8,12,10,16,6,5,17,11,6,0,0,0,11,0,0,0,10,11,13,14,2,0,0,1,0,0,0,15,0,1,17,6,5] >;`

C34.C6 in GAP, Magma, Sage, TeX

`C_3^4.C_6`
`% in TeX`

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

`G:=SmallGroup(486,104);`
`// by ID`

`G=gap.SmallGroup(486,104);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,224,3244,3250,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^6=b,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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