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## G = C34.7S3order 486 = 2·35

### 7th non-split extension by C34 of S3 acting faithfully

Aliases: C34.7S3, C32⋊C95C6, C323(C9⋊C6), C322D95C3, C33.41(C3×S3), C34.C32C2, C33.30(C3⋊S3), (C3×3- 1+2)⋊2S3, C3.8(C33.S3), C32.2(He3⋊C2), (C3×C9)⋊1(C3×S3), C32.37(C3×C3⋊S3), C3.3(C3×He3⋊C2), SmallGroup(486,147)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C34.7S3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C34.C3 — C34.7S3
 Lower central C32⋊C9 — C34.7S3
 Upper central C1 — C3

Generators and relations for C34.7S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=c, ab=ba, ac=ca, ad=da, eae-1=ac-1, af=fa, bc=cb, ebe-1=bd=db, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, df=fd, fef=c-1e2 >

Subgroups: 740 in 147 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C3 [×2], C3 [×10], S3 [×4], C6 [×4], C9 [×6], C32, C32 [×6], C32 [×28], D9 [×3], C3×S3 [×16], C3⋊S3, C3×C6, C3×C9 [×3], C3×C9 [×3], 3- 1+2 [×9], C33 [×2], C33 [×10], C3×D9 [×3], C9⋊C6 [×9], S3×C32 [×4], C3×C3⋊S3 [×4], C32⋊C9, C32⋊C9 [×4], C3×3- 1+2 [×3], C34, C322D9, C3×C9⋊C6 [×3], C32×C3⋊S3, C34.C3, C34.7S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, C9⋊C6 [×3], He3⋊C2 [×3], C3×C3⋊S3, C3×He3⋊C2, C33.S3, C34.7S3

Permutation representations of C34.7S3
On 18 points - transitive group 18T171
Generators in S18
```(1 7 4)(2 5 8)(10 16 13)(11 14 17)
(2 8 5)(3 6 9)(10 16 13)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(9 18)```

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

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

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

`G:=TransitiveGroup(18,171);`

On 27 points - transitive group 27T146
Generators in S27
```(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)
(1 7 4)(2 12 25)(3 20 10)(5 15 19)(6 23 13)(8 18 22)(9 26 16)(11 17 14)(21 27 24)
(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 14 21)(2 15 22)(3 16 23)(4 17 24)(5 18 25)(6 10 26)(7 11 27)(8 12 19)(9 13 20)
(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 18)(11 17)(12 16)(13 15)(19 23)(20 22)(24 27)(25 26)```

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

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

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

`G:=TransitiveGroup(27,146);`

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 He3⋊C2 C9⋊C6 C34.7S3 kernel C34.7S3 C34.C3 C32⋊2D9 C32⋊C9 C3×3- 1+2 C34 C3×C9 C33 C32 C32 C1 # reps 1 1 2 2 3 1 6 2 12 3 6

Matrix representation of C34.7S3 in GL6(𝔽19)

 7 0 0 0 0 12 0 11 0 0 0 1 0 0 1 0 0 0 0 0 0 7 0 12 0 0 0 0 11 1 0 0 0 0 0 1
,
 1 0 0 0 0 7 0 7 0 0 0 0 0 0 11 0 0 8 0 0 0 1 0 7 0 0 0 0 11 8 0 0 0 0 0 7
,
 11 0 0 0 0 8 0 11 0 0 0 8 0 0 11 0 0 8 0 0 0 7 0 0 0 0 0 0 7 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
,
 0 0 11 12 0 8 1 0 0 12 0 7 0 1 0 12 0 7 0 0 0 12 1 7 0 0 0 12 0 7 0 0 0 13 0 7
,
 0 0 18 1 0 0 0 0 18 0 1 0 0 0 18 0 0 0 1 0 18 0 0 0 0 1 18 0 0 0 0 0 10 0 0 1

`G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,12,1,0,12,1,1],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,7,0,8,7,8,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,8,8,8,0,0,7],[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],[0,1,0,0,0,0,0,0,1,0,0,0,11,0,0,0,0,0,12,12,12,12,12,13,0,0,0,1,0,0,8,7,7,7,7,7],[0,0,0,1,0,0,0,0,0,0,1,0,18,18,18,18,18,10,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1] >;`

C34.7S3 in GAP, Magma, Sage, TeX

`C_3^4._7S_3`
`% in TeX`

`G:=Group("C3^4.7S3");`
`// GroupNames label`

`G:=SmallGroup(486,147);`
`// by ID`

`G=gap.SmallGroup(486,147);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,548,338,867,735,3244]);`
`// Polycyclic`

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

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