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## G = C3⋊S3⋊D12order 432 = 24·33

### The semidirect product of C3⋊S3 and D12 acting via D12/C4=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C3⋊S3⋊D12
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C2×C32⋊D6 — C3⋊S3⋊D12
 Lower central He3 — C2×He3 — C3⋊S3⋊D12
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3⋊D12
G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=eae=a-1, ad=da, cbc=b-1, dbd-1=ebe=a-1b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1571 in 205 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4, C4, C22 [×9], S3 [×14], C6, C6 [×9], C2×C4, D4 [×4], C23 [×2], C32 [×2], C32, Dic3 [×2], C12, C12 [×4], D6 [×24], C2×C6 [×5], C2×D4, C3×S3 [×10], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, C4×S3 [×2], D12 [×7], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×6], He3, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C3×C12, S32 [×8], S3×C6 [×9], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×D12, S3×D4 [×2], C32⋊C6 [×2], C32⋊C6 [×2], He3⋊C2 [×2], C2×He3, D6⋊S3 [×2], C3⋊D12 [×2], S3×C12, C3×D12 [×4], C4×C3⋊S3, C12⋊S3, C2×S32 [×4], C32⋊C12, C4×He3, C32⋊D6 [×4], C2×C32⋊C6, C2×C32⋊C6 [×2], C2×He3⋊C2 [×2], S3×D12, D6⋊D6, He33D4 [×2], C4×C32⋊C6, He34D4, He35D4, C2×C32⋊D6 [×2], C3⋊S3⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], S32, C2×D12, S3×D4, C2×S32, C32⋊D6, S3×D12, C2×C32⋊D6, C3⋊S3⋊D12

Smallest permutation representation of C3⋊S3⋊D12
On 36 points
Generators in S36
```(1 36 14)(2 25 15)(3 26 16)(4 27 17)(5 28 18)(6 29 19)(7 30 20)(8 31 21)(9 32 22)(10 33 23)(11 34 24)(12 35 13)
(2 15 25)(3 26 16)(5 18 28)(6 29 19)(8 21 31)(9 32 22)(11 24 34)(12 35 13)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 9 9 18 18 18 18 2 6 6 12 2 18 2 6 6 12 18 18 36 36 36 36 4 6 6 12 12 12 18 18

32 irreducible representations

 dim 1 1 1 1 1 1 12 2 2 2 2 2 2 2 4 4 4 4 6 6 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3⋊S3⋊D12 S3 S3 D4 D6 D6 D6 D12 S32 S3×D4 C2×S32 S3×D12 C32⋊D6 C2×C32⋊D6 kernel C3⋊S3⋊D12 He3⋊3D4 C4×C32⋊C6 He3⋊4D4 He3⋊5D4 C2×C32⋊D6 C1 C4×C3⋊S3 C12⋊S3 C32⋊C6 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C3 C4 C2 # reps 1 2 1 1 1 2 1 1 1 2 1 2 3 4 1 1 1 2 2 2

Matrix representation of C3⋊S3⋊D12 in GL10(𝔽13)

 1 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 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 12 0 12 12 12
,
 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 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 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 12 12 12 0 12 12
,
 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 12 12 12 12
,
 9 0 11 0 0 0 0 0 0 0 0 9 0 11 0 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 12 12 12 12 11 12 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 1 0 0 0 0 0 0 0 1 0 1 0
,
 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 0 0 0 12 12 12 12 11 12 0 0 0 0 0 0 0 0 12 1 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 1 0 0 0 0 0 0 1 0 0 1 0

`G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12],[9,0,2,0,0,0,0,0,0,0,0,9,0,2,0,0,0,0,0,0,11,0,4,0,0,0,0,0,0,0,0,11,0,4,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,11,0,0,1,1,0,0,0,0,1,12,0,0,0,0],[12,0,4,0,0,0,0,0,0,0,0,12,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,11,12,0,0,1,1,0,0,0,0,12,1,0,0,0,0] >;`

C3⋊S3⋊D12 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3\rtimes D_{12}`
`% in TeX`

`G:=Group("C3:S3:D12");`
`// GroupNames label`

`G:=SmallGroup(432,301);`
`// by ID`

`G=gap.SmallGroup(432,301);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,571,4037,537,14118,7069]);`
`// Polycyclic`

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

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