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## G = C12⋊S3order 72 = 23·32

### 1st semidirect product of C12 and S3 acting via S3/C3=C2

Aliases: C121S3, C31D12, C325D4, C6.14D6, C4⋊(C3⋊S3), (C3×C12)⋊1C2, (C3×C6).13C22, (C2×C3⋊S3)⋊2C2, C2.4(C2×C3⋊S3), SmallGroup(72,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C12⋊S3
 Lower central C32 — C3×C6 — C12⋊S3
 Upper central C1 — C2 — C4

Generators and relations for C12⋊S3
G = < a,b,c | a12=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Character table of C12⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 4 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 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 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 linear of order 2 ρ5 2 2 0 0 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ6 2 2 0 0 -1 -1 2 -1 -2 -1 -1 -1 2 -2 1 1 -2 1 1 1 1 orthogonal lifted from D6 ρ7 2 2 0 0 -1 2 -1 -1 -2 -1 -1 2 -1 1 1 1 1 1 1 -2 -2 orthogonal lifted from D6 ρ8 2 2 0 0 2 -1 -1 -1 -2 -1 2 -1 -1 1 1 -2 1 -2 1 1 1 orthogonal lifted from D6 ρ9 2 2 0 0 2 -1 -1 -1 2 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 -1 -1 2 -1 2 -1 -1 -1 2 2 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 0 -1 -1 -1 2 -2 2 -1 -1 -1 1 -2 1 1 1 -2 1 1 orthogonal lifted from D6 ρ12 2 -2 0 0 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 2 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ14 2 -2 0 0 -1 -1 -1 2 0 -2 1 1 1 -√3 0 √3 √3 -√3 0 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 0 0 -1 2 -1 -1 0 1 1 -2 1 √3 √3 √3 -√3 -√3 -√3 0 0 orthogonal lifted from D12 ρ16 2 -2 0 0 2 -1 -1 -1 0 1 -2 1 1 -√3 √3 0 √3 0 -√3 -√3 √3 orthogonal lifted from D12 ρ17 2 -2 0 0 -1 2 -1 -1 0 1 1 -2 1 -√3 -√3 -√3 √3 √3 √3 0 0 orthogonal lifted from D12 ρ18 2 -2 0 0 -1 -1 2 -1 0 1 1 1 -2 0 √3 -√3 0 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ19 2 -2 0 0 -1 -1 2 -1 0 1 1 1 -2 0 -√3 √3 0 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ20 2 -2 0 0 2 -1 -1 -1 0 1 -2 1 1 √3 -√3 0 -√3 0 √3 √3 -√3 orthogonal lifted from D12 ρ21 2 -2 0 0 -1 -1 -1 2 0 -2 1 1 1 √3 0 -√3 -√3 √3 0 -√3 √3 orthogonal lifted from D12

Smallest permutation representation of C12⋊S3
On 36 points
Generators in S36
```(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 13)(2 36 14)(3 25 15)(4 26 16)(5 27 17)(6 28 18)(7 29 19)(8 30 20)(9 31 21)(10 32 22)(11 33 23)(12 34 24)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)```

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

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

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

C12⋊S3 is a maximal subgroup of
C3⋊D24  C325SD16  C242S3  C325D8  C327D8  C3211SD16  D6.6D6  S3×D12  C12.59D6  D4×C3⋊S3  C12.26D6  He34D4  C36⋊S3  C338D4  C3312D4  C12⋊S4  C12.7S4  C15⋊D12  C60⋊S3
C12⋊S3 is a maximal quotient of
C242S3  C325D8  C325Q16  C12⋊Dic3  C6.11D12  C36⋊S3  He35D4  C338D4  C3312D4  C12⋊S4  C15⋊D12  C60⋊S3

Matrix representation of C12⋊S3 in GL4(𝔽13) generated by

 0 1 0 0 12 12 0 0 0 0 6 3 0 0 10 3
,
 1 0 0 0 0 1 0 0 0 0 12 12 0 0 1 0
,
 1 0 0 0 12 12 0 0 0 0 12 12 0 0 0 1
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,6,10,0,0,3,3],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,12,1] >;`

C12⋊S3 in GAP, Magma, Sage, TeX

`C_{12}\rtimes S_3`
`% in TeX`

`G:=Group("C12:S3");`
`// GroupNames label`

`G:=SmallGroup(72,33);`
`// by ID`

`G=gap.SmallGroup(72,33);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,323,1204]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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