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## G = C6.S32order 216 = 23·33

### 2nd non-split extension by C6 of S32 acting via S32/C3⋊S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C6.S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32
 Lower central He3 — C6.S32
 Upper central C1 — C2

Generators and relations for C6.S32
G = < a,b,c,d | a3=b3=c6=d4=1, dad-1=ab=ba, cac-1=a-1b-1, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 282 in 62 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, S3×Dic3, C6.D6, C32⋊C12, He33C4, C2×C32⋊C6, C6.S32
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C4×S3, C2×Dic3, S32, S3×Dic3, C32⋊D6, C6.S32

Character table of C6.S32

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D size 1 1 9 9 2 6 6 12 9 9 9 9 2 6 6 12 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 trivial ρ2 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 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 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 ρ5 1 -1 -1 1 1 1 1 1 -i -i i i -1 -1 -1 -1 1 -1 -i i -i i linear of order 4 ρ6 1 -1 1 -1 1 1 1 1 -i i i -i -1 -1 -1 -1 -1 1 -i i i -i linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 i -i -i i -1 -1 -1 -1 -1 1 i -i -i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 i i -i -i -1 -1 -1 -1 1 -1 i -i i -i linear of order 4 ρ9 2 2 0 0 2 -1 2 -1 0 -2 0 -2 2 2 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 2 2 -1 -1 0 0 0 0 2 -1 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ11 2 2 0 0 2 -1 2 -1 0 2 0 2 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 -2 2 2 -1 -1 0 0 0 0 2 -1 2 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 2 -1 -1 0 0 0 0 -2 1 -2 1 -1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ14 2 -2 2 -2 2 2 -1 -1 0 0 0 0 -2 1 -2 1 1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 0 0 2 -1 2 -1 0 2i 0 -2i -2 -2 1 1 0 0 0 0 -i i complex lifted from C4×S3 ρ16 2 -2 0 0 2 -1 2 -1 0 -2i 0 2i -2 -2 1 1 0 0 0 0 i -i complex lifted from C4×S3 ρ17 4 4 0 0 4 -2 -2 1 0 0 0 0 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ18 4 -4 0 0 4 -2 -2 1 0 0 0 0 -4 2 2 -1 0 0 0 0 0 0 symplectic lifted from S3×Dic3, Schur index 2 ρ19 6 6 0 0 -3 0 0 0 -2 0 -2 0 -3 0 0 0 0 0 1 1 0 0 orthogonal lifted from C32⋊D6 ρ20 6 6 0 0 -3 0 0 0 2 0 2 0 -3 0 0 0 0 0 -1 -1 0 0 orthogonal lifted from C32⋊D6 ρ21 6 -6 0 0 -3 0 0 0 -2i 0 2i 0 3 0 0 0 0 0 i -i 0 0 complex faithful ρ22 6 -6 0 0 -3 0 0 0 2i 0 -2i 0 3 0 0 0 0 0 -i i 0 0 complex faithful

Smallest permutation representation of C6.S32
On 36 points
Generators in S36
```(1 28 19)(2 29 20)(4 22 25)(5 23 26)(8 32 17)(9 33 18)(11 14 35)(12 15 36)
(1 28 19)(2 20 29)(3 30 21)(4 22 25)(5 26 23)(6 24 27)(7 16 31)(8 32 17)(9 18 33)(10 34 13)(11 14 35)(12 36 15)
(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 28 29 30)(31 32 33 34 35 36)
(1 35 4 32)(2 34 5 31)(3 33 6 36)(7 29 13 23)(8 28 14 22)(9 27 15 21)(10 26 16 20)(11 25 17 19)(12 30 18 24)```

`G:=sub<Sym(36)| (1,28,19)(2,29,20)(4,22,25)(5,23,26)(8,32,17)(9,33,18)(11,14,35)(12,15,36), (1,28,19)(2,20,29)(3,30,21)(4,22,25)(5,26,23)(6,24,27)(7,16,31)(8,32,17)(9,18,33)(10,34,13)(11,14,35)(12,36,15), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,29,13,23)(8,28,14,22)(9,27,15,21)(10,26,16,20)(11,25,17,19)(12,30,18,24)>;`

`G:=Group( (1,28,19)(2,29,20)(4,22,25)(5,23,26)(8,32,17)(9,33,18)(11,14,35)(12,15,36), (1,28,19)(2,20,29)(3,30,21)(4,22,25)(5,26,23)(6,24,27)(7,16,31)(8,32,17)(9,18,33)(10,34,13)(11,14,35)(12,36,15), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,29,13,23)(8,28,14,22)(9,27,15,21)(10,26,16,20)(11,25,17,19)(12,30,18,24) );`

`G=PermutationGroup([[(1,28,19),(2,29,20),(4,22,25),(5,23,26),(8,32,17),(9,33,18),(11,14,35),(12,15,36)], [(1,28,19),(2,20,29),(3,30,21),(4,22,25),(5,26,23),(6,24,27),(7,16,31),(8,32,17),(9,18,33),(10,34,13),(11,14,35),(12,36,15)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,35,4,32),(2,34,5,31),(3,33,6,36),(7,29,13,23),(8,28,14,22),(9,27,15,21),(10,26,16,20),(11,25,17,19),(12,30,18,24)]])`

C6.S32 is a maximal subgroup of   C3⋊S3⋊Dic6  C12⋊S3⋊S3  C12.84S32  C4×C32⋊D6  C62.8D6  C62.9D6  C62⋊D6
C6.S32 is a maximal quotient of   C32⋊C6⋊C8  He3⋊M4(2)  He3⋊C42  C62.D6  C62.4D6

Matrix representation of C6.S32 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 12 0 0 0 0 12 0 0 0 0 0
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8 0 0 0

`G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,0,0,0,0,12,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0] >;`

C6.S32 in GAP, Magma, Sage, TeX

`C_6.S_3^2`
`% in TeX`

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

`G:=SmallGroup(216,34);`
`// by ID`

`G=gap.SmallGroup(216,34);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,201,1444,382,5189,2603]);`
`// Polycyclic`

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

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