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## G = C3≀C3⋊S3order 486 = 2·35

### 2nd semidirect product of C3≀C3 and S3 acting via S3/C3=C2

Aliases: C3≀C32S3, C9○He32S3, (C32×C9)⋊21S3, He3.C32S3, C9.He32C2, He3⋊C36S3, He3.5(C3⋊S3), C3.He32S3, C33.41(C3⋊S3), C9.5(He3⋊C2), C3.7(He35S3), C32.5(C33⋊C2), 3- 1+2.2(C3⋊S3), (C3×C9).14(C3⋊S3), SmallGroup(486,189)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9.He3 — C3≀C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C9.He3 — C3≀C3⋊S3
 Lower central C9.He3 — C3≀C3⋊S3
 Upper central C1

Generators and relations for C3≀C3⋊S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, eae-1=ab=ba, cac-1=faf=ab-1, ede-1=ad=da, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=ab-1c, ece-1=a-1c, fcf=a-1c-1, fdf=d-1, fef=e-1 >

Subgroups: 1060 in 119 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C33⋊S3, He3.3S3, He3⋊S3, 3- 1+2.S3, C3×C9⋊S3, He3.4S3, C9.He3, C3≀C3⋊S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, He35S3, C3≀C3⋊S3

Character table of C3≀C3⋊S3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O 9P 9Q size 1 81 2 3 3 6 6 6 18 18 18 81 81 2 2 2 6 6 6 6 6 6 6 6 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 2 2 2 -1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 -1 2 2 2 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 -1 -1 -1 -1 2 -1 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 2 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 -1 -1 -1 -1 2 -1 0 0 2 2 2 2 -1 -1 -1 -1 2 -1 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ6 2 0 2 2 2 -1 -1 -1 2 -1 -1 0 0 2 2 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 orthogonal lifted from S3 ρ7 2 0 2 2 2 -1 -1 -1 -1 -1 2 0 0 2 2 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 2 2 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 0 2 2 2 -1 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 2 2 2 2 2 -1 2 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ11 2 0 2 2 2 2 2 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 2 orthogonal lifted from S3 ρ12 2 0 2 2 2 -1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 2 orthogonal lifted from S3 ρ13 2 0 2 2 2 2 2 2 -1 -1 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 2 -1 orthogonal lifted from S3 ρ14 2 0 2 2 2 -1 -1 -1 2 2 2 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 0 2 2 2 -1 -1 -1 -1 -1 2 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 2 -1 -1 2 orthogonal lifted from S3 ρ16 3 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ6 ζ65 3 3 3 -3+3√-3/2 0 0 0 0 -3-3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ17 3 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ65 ζ6 3 3 3 -3-3√-3/2 0 0 0 0 -3+3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ18 3 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ32 ζ3 3 3 3 -3+3√-3/2 0 0 0 0 -3-3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ19 3 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ3 ζ32 3 3 3 -3-3√-3/2 0 0 0 0 -3+3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ20 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 ζ95+2ζ94-ζ92+ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ21 6 0 -3 0 0 0 3 -3 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 -ζ98+2ζ97+ζ94+ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ22 6 0 -3 0 0 -3 0 3 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 -ζ98+2ζ97+ζ94+ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ23 6 0 -3 0 0 -3 0 3 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 2ζ98-ζ94+ζ92+ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ24 6 0 -3 0 0 0 3 -3 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 2ζ98-ζ94+ζ92+ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ25 6 0 -3 0 0 0 3 -3 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 ζ95+2ζ94-ζ92+ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ26 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 -ζ98+2ζ97+ζ94+ζ92 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ27 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 2ζ98-ζ94+ζ92+ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ28 6 0 -3 0 0 -3 0 3 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 ζ95+2ζ94-ζ92+ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ29 6 0 6 -3+3√-3 -3-3√-3 0 0 0 0 0 0 0 0 -3 -3 -3 3-3√-3/2 0 0 0 0 3+3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊5S3 ρ30 6 0 6 -3-3√-3 -3+3√-3 0 0 0 0 0 0 0 0 -3 -3 -3 3+3√-3/2 0 0 0 0 3-3√-3/2 0 0 0 0 0 0 0 0 complex lifted from He3⋊5S3

Permutation representations of C3≀C3⋊S3
On 27 points - transitive group 27T159
Generators in S27
```(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 17 18)(19 21 20)(22 23 24)(25 26 27)
(1 14 22)(2 13 23)(3 15 24)(4 11 16)(5 10 17)(6 12 18)(7 21 25)(8 19 27)(9 20 26)
(1 3 2)(4 6 5)(7 8 9)(16 18 17)(22 24 23)(25 27 26)
(1 26 11)(2 27 10)(3 25 12)(4 22 21)(5 23 20)(6 24 19)(7 16 15)(8 18 13)(9 17 14)
(1 16)(2 18)(3 17)(4 22)(5 24)(6 23)(7 26)(8 27)(9 25)(10 13)(11 15)(12 14)(19 20)```

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

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

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

`G:=TransitiveGroup(27,159);`

Matrix representation of C3≀C3⋊S3 in GL6(𝔽19)

 0 1 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 1 0
,
 18 18 0 0 0 0 1 0 0 0 0 0 0 0 18 18 0 0 0 0 1 0 0 0 0 0 0 0 18 18 0 0 0 0 1 0
,
 0 0 5 7 0 0 0 0 12 17 0 0 0 0 0 0 5 7 0 0 0 0 12 17 12 17 0 0 0 0 2 14 0 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 18 18
,
 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 17 5 0 0 0 0 7 2 0 0 0 0 0 0 0 0 17 5 0 0 0 0 7 2 0 0 17 5 0 0 0 0 7 2 0 0

`G:=sub<GL(6,GF(19))| [0,18,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[0,0,0,0,12,2,0,0,0,0,17,14,5,12,0,0,0,0,7,17,0,0,0,0,0,0,5,12,0,0,0,0,7,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[17,7,0,0,0,0,5,2,0,0,0,0,0,0,0,0,17,7,0,0,0,0,5,2,0,0,17,7,0,0,0,0,5,2,0,0] >;`

C3≀C3⋊S3 in GAP, Magma, Sage, TeX

`C_3\wr C_3\rtimes S_3`
`% in TeX`

`G:=Group("C3wrC3:S3");`
`// GroupNames label`

`G:=SmallGroup(486,189);`
`// by ID`

`G=gap.SmallGroup(486,189);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1993,1951,218,867,303,11344,1096,11669]);`
`// Polycyclic`

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

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