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G = C3≀C3.C6order 486 = 2·35

The non-split extension by C3≀C3 of C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3≀C3.C6
 Chief series C1 — C3 — C32 — He3 — C9○He3 — C9.2He3 — C3≀C3.C6
 Lower central He3 — C3≀C3.C6
 Upper central C1 — C3 — C9

Generators and relations for C3≀C3.C6
G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=b-1, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1b, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ece-1=c-1 >

Subgroups: 306 in 69 conjugacy classes, 21 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, C3≀C3, C3≀C3, He3.C3, He3.C3, He3⋊C3, C3.He3, C3×3- 1+2, C9○He3, C9○He3, C3≀S3, He3.C6, He3.2C6, S3×3- 1+2, He3.4C6, C9.2He3, C3≀C3.C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C3≀C3.C6

Permutation representations of C3≀C3.C6
On 27 points - transitive group 27T201
Generators in S27
```(1 7 4)(2 8 5)(3 9 6)(10 22 16)(12 24 18)(14 26 20)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 10 19)(2 20 11)(3 12 21)(4 22 13)(5 14 23)(6 24 15)(7 16 25)(8 26 17)(9 18 27)
(2 5 8)(3 9 6)(10 16 22)(12 24 18)(13 19 25)(15 27 21)
(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)(2,8,5)(3,9,6)(10,22,16)(12,24,18)(14,26,20), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (2,5,8)(3,9,6)(10,16,22)(12,24,18)(13,19,25)(15,27,21), (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)(2,8,5)(3,9,6)(10,22,16)(12,24,18)(14,26,20), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (2,5,8)(3,9,6)(10,16,22)(12,24,18)(13,19,25)(15,27,21), (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),(2,8,5),(3,9,6),(10,22,16),(12,24,18),(14,26,20)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,10,19),(2,20,11),(3,12,21),(4,22,13),(5,14,23),(6,24,15),(7,16,25),(8,26,17),(9,18,27)], [(2,5,8),(3,9,6),(10,16,22),(12,24,18),(13,19,25),(15,27,21)], [(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(27,201);`

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 6 6 9 type + + + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 S3 C3×S3 C3×S3 C32⋊C6 C3×C32⋊C6 C3≀C3.C6 kernel C3≀C3.C6 C9.2He3 C3≀S3 He3.C6 He3.2C6 He3.4C6 C3≀C3 He3.C3 He3⋊C3 C9○He3 C3×3- 1+2 C3×C9 C33 C9 C3 C1 # reps 1 1 2 2 2 2 2 2 2 2 1 6 2 1 2 4

Matrix representation of C3≀C3.C6 in GL9(𝔽19)

 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 1 12 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 1 12 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7
,
 12 12 13 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 8 7 7 0 0 0 0 0 0 0 0 12 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 8 7 7 0 1 0 0 0 0 0 0 12 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 8 7 7 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 18 0 11 0 0 0 0 0 0 12 0 0 7 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 11 0 0 0 11 0 0 0 0 0 7 0 7 8 0 0 0 0 0 0 7
,
 11 0 0 0 0 0 15 0 0 8 8 0 0 0 0 0 0 15 0 0 0 0 0 0 11 11 11 11 0 0 0 0 0 8 0 0 8 8 7 0 0 0 0 0 8 0 11 0 0 0 0 11 0 11 11 0 0 7 0 0 8 0 0 8 8 0 0 0 7 0 0 8 0 11 0 0 7 0 11 0 11

`G:=sub<GL(9,GF(19))| [11,0,1,0,0,1,0,0,1,0,7,12,0,0,12,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[12,11,8,0,0,8,0,0,8,12,0,7,0,0,7,0,0,7,13,0,7,12,0,7,12,0,7,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0],[1,0,18,12,0,0,1,0,7,0,11,0,0,18,1,0,11,8,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,8,0,11,8,0,11,8,0,0,8,0,0,8,11,0,8,11,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,15,0,11,8,0,11,8,0,11,0,0,11,0,0,0,0,0,0,0,15,11,0,8,11,0,8,11] >;`

C3≀C3.C6 in GAP, Magma, Sage, TeX

`C_3\wr C_3.C_6`
`% in TeX`

`G:=Group("C3wrC3.C6");`
`// GroupNames label`

`G:=SmallGroup(486,132);`
`// by ID`

`G=gap.SmallGroup(486,132);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,500,867,873,8104,382]);`
`// Polycyclic`

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

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