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## G = C62⋊S3order 216 = 23·33

### 4th semidirect product of C62 and S3 acting faithfully

Aliases: C624S3, C321S4, C3⋊S4⋊C3, (C3×A4)⋊C6, C3.3(C3×S4), C32⋊A41C2, C22⋊(C32⋊C6), (C2×C6).4(C3×S3), SmallGroup(216,92)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C62⋊S3
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32⋊A4 — C62⋊S3
 Lower central C3×A4 — C62⋊S3
 Upper central C1

Generators and relations for C62⋊S3
G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

3C2
18C2
3C3
12C3
24C3
9C4
9C22
3C6
3C6
6C6
6S3
18C6
36S3
4C32
8C32
9D4
3D6
3Dic3
3A4
6A4
9C12
4He3
9S4

Character table of C62⋊S3

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 3 18 2 3 3 24 24 24 18 3 3 6 6 6 18 18 18 18 ρ1 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 linear of order 2 ρ3 1 1 -1 1 ζ3 ζ32 ζ3 ζ32 1 -1 ζ3 ζ32 ζ32 1 ζ3 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 -1 1 ζ32 ζ3 ζ32 ζ3 1 -1 ζ32 ζ3 ζ3 1 ζ32 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ7 2 2 0 2 2 2 -1 -1 -1 0 2 2 2 2 2 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 -1+√-3 -1-√-3 -1-√-3 2 -1+√-3 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 -1-√-3 -1+√-3 -1+√-3 2 -1-√-3 0 0 0 0 complex lifted from C3×S3 ρ10 3 -1 -1 3 3 3 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from S4 ρ11 3 -1 1 3 3 3 0 0 0 -1 -1 -1 -1 -1 -1 1 1 -1 -1 orthogonal lifted from S4 ρ12 3 -1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 ζ65 ζ6 ζ6 -1 ζ65 ζ32 ζ3 ζ6 ζ65 complex lifted from C3×S4 ρ13 3 -1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 ζ6 ζ65 ζ65 -1 ζ6 ζ3 ζ32 ζ65 ζ6 complex lifted from C3×S4 ρ14 3 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 1 ζ65 ζ6 ζ6 -1 ζ65 ζ6 ζ65 ζ32 ζ3 complex lifted from C3×S4 ρ15 3 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 1 ζ6 ζ65 ζ65 -1 ζ6 ζ65 ζ6 ζ3 ζ32 complex lifted from C3×S4 ρ16 6 -2 0 -3 0 0 0 0 0 0 4 4 -2 1 -2 0 0 0 0 orthogonal faithful ρ17 6 6 0 -3 0 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ18 6 -2 0 -3 0 0 0 0 0 0 -2+2√-3 -2-2√-3 1+√-3 1 1-√-3 0 0 0 0 complex faithful ρ19 6 -2 0 -3 0 0 0 0 0 0 -2-2√-3 -2+2√-3 1-√-3 1 1+√-3 0 0 0 0 complex faithful

Permutation representations of C62⋊S3
On 18 points - transitive group 18T97
Generators in S18
```(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 8 9)(10 12 11)(13 15 14)(16 17 18)
(1 2)(3 4)(5 6)(10 15)(11 13)(12 14)
(7 16)(8 17)(9 18)(10 15)(11 13)(12 14)
(1 14 8)(2 12 17)(3 15 7)(4 10 16)(5 13 9)(6 11 18)
(1 2)(3 6)(4 5)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)```

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

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

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

`G:=TransitiveGroup(18,97);`

On 18 points - transitive group 18T99
Generators in S18
```(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 9 8)(10 11 12)(13 15 14)(16 17 18)
(1 2)(3 4)(5 6)(7 14)(8 15)(9 13)
(7 14)(8 15)(9 13)(10 16)(11 17)(12 18)
(1 8 16)(2 15 10)(3 9 18)(4 13 12)(5 7 17)(6 14 11)
(3 5)(4 6)(7 18)(8 16)(9 17)(10 15)(11 13)(12 14)```

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

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

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

`G:=TransitiveGroup(18,99);`

C62⋊S3 is a maximal subgroup of   C625D6
C62⋊S3 is a maximal quotient of   C32⋊CSU2(𝔽3)  C322GL2(𝔽3)  C625Dic3

Matrix representation of C62⋊S3 in GL6(ℤ)

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

`G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0] >;`

C62⋊S3 in GAP, Magma, Sage, TeX

`C_6^2\rtimes S_3`
`% in TeX`

`G:=Group("C6^2:S3");`
`// GroupNames label`

`G:=SmallGroup(216,92);`
`// by ID`

`G=gap.SmallGroup(216,92);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-2,2,218,224,867,3244,556,1949,989]);`
`// Polycyclic`

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

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