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## G = C34.S3order 486 = 2·35

### 4th non-split extension by C34 of S3 acting faithfully

Aliases: C34.4S3, C32⋊C94C6, C9⋊S31C32, C32⋊D9⋊C3, C322(C9⋊C6), C33.34(C3×S3), C34.C31C2, C33.S31C3, C32.6(C32⋊C6), C32.38(S3×C32), (C3×3- 1+2)⋊1C6, (C3×C9)⋊1(C3×C6), C3.2(C3×C9⋊C6), C3.3(C3×C32⋊C6), SmallGroup(486,105)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C34.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C34.C3 — C34.S3
 Lower central C3×C9 — C34.S3
 Upper central C1

Generators and relations for C34.S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=d, ab=ba, ac=ca, ad=da, eae-1=ac-1, af=fa, bc=cb, bd=db, ebe-1=bd-1, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=d-1e2 >

Subgroups: 708 in 129 conjugacy classes, 27 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, 3- 1+2, C33, C33, C33, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C3×3- 1+2, C3×3- 1+2, C34, C32⋊D9, C33.S3, C32×C3⋊S3, C34.C3, C34.S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, C9⋊C6, S3×C32, C3×C32⋊C6, C3×C9⋊C6, C34.S3

Permutation representations of C34.S3
On 27 points - transitive group 27T198
Generators in S27
```(2 20 15)(3 16 21)(5 23 18)(6 10 24)(8 26 12)(9 13 27)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(20 26 23)(21 24 27)
(1 14 19)(2 15 20)(3 16 21)(4 17 22)(5 18 23)(6 10 24)(7 11 25)(8 12 26)(9 13 27)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(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)
(2 9)(3 8)(4 7)(5 6)(10 23)(11 22)(12 21)(13 20)(14 19)(15 27)(16 26)(17 25)(18 24)```

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

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

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

`G:=TransitiveGroup(27,198);`

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3L 3M ··· 3T 6A ··· 6H 9A ··· 9I order 1 2 3 3 3 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 2 2 2 2 3 ··· 3 6 ··· 6 27 ··· 27 18 ··· 18

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C32⋊C6 C9⋊C6 C3×C32⋊C6 C3×C9⋊C6 kernel C34.S3 C34.C3 C32⋊D9 C33.S3 C32⋊C9 C3×3- 1+2 C34 C33 C32 C32 C3 C3 # reps 1 1 6 2 6 2 1 8 1 3 2 6

Matrix representation of C34.S3 in GL12(ℤ)

 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0
,
 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1
,
 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 1 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0],[0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0] >;`

C34.S3 in GAP, Magma, Sage, TeX

`C_3^4.S_3`
`% in TeX`

`G:=Group("C3^4.S3");`
`// GroupNames label`

`G:=SmallGroup(486,105);`
`// by ID`

`G=gap.SmallGroup(486,105);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,873,735,453,3244,11669]);`
`// Polycyclic`

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

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