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## G = C3≀C3.S3order 486 = 2·35

### The non-split extension by C3≀C3 of S3 acting via S3/C3=C2

Aliases: C3≀C3.S3, C9○He31S3, C9.(C32⋊C6), (C32×C9)⋊17C6, He3.C34S3, C9.He31C2, He3⋊C34S3, C324D98C3, He3.3(C3⋊S3), C33.67(C3×S3), C3.9(He34S3), (C3×C9).35(C3×S3), C32.19(C3×C3⋊S3), SmallGroup(486,175)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3≀C3.S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=ab-1c, ce=ec, cf=fc, de=ed, fdf=d-1, fef=b-1e2 >

Subgroups: 980 in 86 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C33⋊C2, C3≀C3, C3≀C3, He3.C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C9○He3, C33⋊C6, He3.S3, He3.2S3, He3.4S3, C324D9, C9.He3, C3≀C3.S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He34S3, 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 6 6 6 6 9 9 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 1 -1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 6 ρ4 1 -1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 6 ρ5 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 2 0 2 2 2 2 2 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ8 2 0 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ9 2 0 2 2 -1 -1 -1 2 2 -1 -1 0 0 2 2 2 -1 2 2 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ10 2 0 2 2 -1 -1 -1 2 2 2 2 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 0 2 2 -1 -1 -1 -1+√-3 -1-√-3 -1+√-3 -1-√-3 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×S3 ρ12 2 0 2 2 -1 -1 -1 -1-√-3 -1+√-3 -1-√-3 -1+√-3 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×S3 ρ13 2 0 2 2 2 2 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1-√-3 ζ65 ζ6 ζ65 -1+√-3 ζ6 complex lifted from C3×S3 ρ14 2 0 2 2 -1 -1 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 ζ65 ζ6 -1+√-3 -1-√-3 ζ6 ζ65 complex lifted from C3×S3 ρ15 2 0 2 2 -1 -1 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 2 -1 2 2 -1 -1 -1 -1 -1 ζ6 -1+√-3 ζ6 ζ65 ζ65 -1-√-3 complex lifted from C3×S3 ρ16 2 0 2 2 -1 -1 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 2 -1 2 2 -1 -1 -1 -1 -1 ζ65 -1-√-3 ζ65 ζ6 ζ6 -1+√-3 complex lifted from C3×S3 ρ17 2 0 2 2 2 2 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1+√-3 ζ6 ζ65 ζ6 -1-√-3 ζ65 complex lifted from C3×S3 ρ18 2 0 2 2 -1 -1 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 ζ6 ζ65 -1-√-3 -1+√-3 ζ65 ζ6 complex lifted from C3×S3 ρ19 6 0 6 -3 0 0 0 0 0 0 0 0 0 6 6 6 0 -3 -3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 0 6 -3 0 0 0 0 0 0 0 0 0 -3 -3 -3 0 -3 6 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ21 6 0 6 -3 0 0 0 0 0 0 0 0 0 -3 -3 -3 0 6 -3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ22 6 0 -3 0 -3 0 3 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ23 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 2ζ98-ζ94+ζ92+ζ9 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ24 6 0 -3 0 3 -3 0 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 ζ95+2ζ94-ζ92+ζ9 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ25 6 0 -3 0 3 -3 0 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ26 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 ζ95+2ζ94-ζ92+ζ9 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal faithful ρ27 6 0 -3 0 3 -3 0 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 -ζ98+2ζ97+ζ94+ζ92 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ28 6 0 -3 0 -3 0 3 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ29 6 0 -3 0 -3 0 3 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 2ζ98-ζ94+ζ92+ζ9 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful ρ30 6 0 -3 0 0 3 -3 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 -ζ98+2ζ97+ζ94+ζ92 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C3≀C3.S3
On 27 points - transitive group 27T170
Generators in S27
```(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(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 22 13)(2 23 14)(3 24 15)(4 25 16)(5 26 17)(6 27 18)(7 19 10)(8 20 11)(9 21 12)
(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 16)(11 15)(12 14)(17 18)(19 25)(20 24)(21 23)(26 27)```

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

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

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

`G:=TransitiveGroup(27,170);`

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

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

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

C3≀C3.S3 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(486,175);`
`// by ID`

`G=gap.SmallGroup(486,175);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,4755,453,3244,3250,11669]);`
`// Polycyclic`

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

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