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## G = C9⋊C9.3S3order 486 = 2·35

### 3rd non-split extension by C9⋊C9 of S3 acting faithfully

Aliases: C9⋊C9.3S3, C32.5He3⋊C2, C3.He3.3C6, C32.4(C32⋊C6), C3.8(He3.2C6), 3- 1+2.S3.2C3, (C3×C9).3(C3×S3), SmallGroup(486,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3.He3 — C9⋊C9.3S3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C32.5He3 — C9⋊C9.3S3
 Lower central C3.He3 — C9⋊C9.3S3
 Upper central C1

Generators and relations for C9⋊C9.3S3
G = < a,b,c,d | a9=b9=d2=1, c3=a3, bab-1=a7, cac-1=a4b6, dad=a-1, cbc-1=a7b4, bd=db, dcd=a6c2 >

Character table of C9⋊C9.3S3

 class 1 2 3A 3B 3C 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 18A 18B 18C 18D 18E 18F size 1 27 2 3 3 27 27 9 9 9 9 9 9 18 54 54 54 27 27 27 27 27 27 ρ1 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 linear of order 2 ρ3 1 -1 1 1 1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ5 1 -1 1 1 1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ7 2 0 2 2 2 0 0 2 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 2 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 2 -1 ζ65 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ9 2 0 2 2 2 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 2 -1 ζ6 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ10 3 -1 3 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 ζ95+2ζ92 2ζ94+ζ9 ζ98+2ζ95 2ζ97+ζ94 ζ97+2ζ9 2ζ98+ζ92 0 0 0 0 -ζ98 -ζ94 -ζ97 -ζ92 -ζ95 -ζ9 complex lifted from He3.2C6 ρ11 3 -1 3 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 ζ98+2ζ95 ζ97+2ζ9 2ζ98+ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ95+2ζ92 0 0 0 0 -ζ92 -ζ9 -ζ94 -ζ95 -ζ98 -ζ97 complex lifted from He3.2C6 ρ12 3 1 3 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 2ζ97+ζ94 ζ98+2ζ95 2ζ94+ζ9 ζ95+2ζ92 2ζ98+ζ92 ζ97+2ζ9 0 0 0 0 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 complex lifted from He3.2C6 ρ13 3 -1 3 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 2ζ94+ζ9 2ζ98+ζ92 ζ97+2ζ9 ζ98+2ζ95 ζ95+2ζ92 2ζ97+ζ94 0 0 0 0 -ζ97 -ζ98 -ζ95 -ζ94 -ζ9 -ζ92 complex lifted from He3.2C6 ρ14 3 1 3 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 2ζ98+ζ92 2ζ97+ζ94 ζ95+2ζ92 ζ97+2ζ9 2ζ94+ζ9 ζ98+2ζ95 0 0 0 0 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 complex lifted from He3.2C6 ρ15 3 1 3 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 2ζ94+ζ9 2ζ98+ζ92 ζ97+2ζ9 ζ98+2ζ95 ζ95+2ζ92 2ζ97+ζ94 0 0 0 0 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 complex lifted from He3.2C6 ρ16 3 -1 3 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 2ζ98+ζ92 2ζ97+ζ94 ζ95+2ζ92 ζ97+2ζ9 2ζ94+ζ9 ζ98+2ζ95 0 0 0 0 -ζ95 -ζ97 -ζ9 -ζ98 -ζ92 -ζ94 complex lifted from He3.2C6 ρ17 3 -1 3 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 ζ97+2ζ9 ζ95+2ζ92 2ζ97+ζ94 2ζ98+ζ92 ζ98+2ζ95 2ζ94+ζ9 0 0 0 0 -ζ94 -ζ92 -ζ98 -ζ9 -ζ97 -ζ95 complex lifted from He3.2C6 ρ18 3 -1 3 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 2ζ97+ζ94 ζ98+2ζ95 2ζ94+ζ9 ζ95+2ζ92 2ζ98+ζ92 ζ97+2ζ9 0 0 0 0 -ζ9 -ζ95 -ζ92 -ζ97 -ζ94 -ζ98 complex lifted from He3.2C6 ρ19 3 1 3 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 ζ97+2ζ9 ζ95+2ζ92 2ζ97+ζ94 2ζ98+ζ92 ζ98+2ζ95 2ζ94+ζ9 0 0 0 0 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 complex lifted from He3.2C6 ρ20 3 1 3 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 ζ95+2ζ92 2ζ94+ζ9 ζ98+2ζ95 2ζ97+ζ94 ζ97+2ζ9 2ζ98+ζ92 0 0 0 0 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 complex lifted from He3.2C6 ρ21 3 1 3 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 ζ98+2ζ95 ζ97+2ζ9 2ζ98+ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ95+2ζ92 0 0 0 0 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 complex lifted from He3.2C6 ρ22 6 0 6 6 6 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ23 18 0 -9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C9⋊C9.3S3
On 27 points - transitive group 27T209
Generators in S27
```(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 5 8)(3 9 6)(10 14 12 13 17 15 16 11 18)(19 27 23 25 24 20 22 21 26)
(1 17 26 4 11 20 7 14 23)(2 15 24 5 18 27 8 12 21)(3 13 22 6 16 25 9 10 19)
(2 9)(3 8)(4 7)(5 6)(10 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)```

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

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

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

`G:=TransitiveGroup(27,209);`

Matrix representation of C9⋊C9.3S3 in GL18(ℤ)

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

`G:=sub<GL(18,Integers())| [0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0] >;`

C9⋊C9.3S3 in GAP, Magma, Sage, TeX

`C_9\rtimes C_9._3S_3`
`% in TeX`

`G:=Group("C9:C9.3S3");`
`// GroupNames label`

`G:=SmallGroup(486,40);`
`// by ID`

`G=gap.SmallGroup(486,40);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,6050,224,500,6051,2169,951,453,1096,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^9=d^2=1,c^3=a^3,b*a*b^-1=a^7,c*a*c^-1=a^4*b^6,d*a*d=a^-1,c*b*c^-1=a^7*b^4,b*d=d*b,d*c*d=a^6*c^2>;`
`// generators/relations`

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