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## G = C12⋊3S32order 432 = 24·33

### 3rd semidirect product of C12 and S32 acting via S32/C32=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C12⋊3S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C2×C32⋊4D6 — C12⋊3S32
 Lower central C33 — C32×C6 — C12⋊3S32
 Upper central C1 — C2 — C4

Generators and relations for C123S32
G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1752 in 270 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4, C4, C22 [×9], S3 [×18], C6, C6 [×2], C6 [×10], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×3], C12, C12 [×2], C12 [×5], D6 [×27], C2×C6 [×5], C2×D4, C3×S3 [×18], C3⋊S3 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×3], D12 [×8], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×6], C33, C3×Dic3 [×3], C3⋊Dic3, C3×C12, C3×C12 [×2], C3×C12 [×4], S32 [×12], S3×C6 [×15], C2×C3⋊S3, C2×C3⋊S3 [×4], C2×D12, S3×D4 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3 [×4], C32×C6, D6⋊S3 [×2], C3⋊D12 [×4], S3×C12 [×3], C3×D12 [×6], C4×C3⋊S3, C12⋊S3 [×2], C2×S32 [×6], C3×C3⋊Dic3, C32×C12, C324D6 [×4], C6×C3⋊S3, C6×C3⋊S3 [×4], S3×D12 [×2], D6⋊D6, C339D4 [×2], C12×C3⋊S3, C3×C12⋊S3 [×2], C2×C324D6 [×2], C123S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, D12 [×2], C22×S3 [×3], S32 [×3], C2×D12, S3×D4 [×2], C2×S32 [×3], C324D6, S3×D12 [×2], D6⋊D6, C2×C324D6, C123S32

Smallest permutation representation of C123S32
On 48 points
Generators in S48
```(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(25 37)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)```

`G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,35)(2,36)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)>;`

`G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,35)(2,36)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38) );`

`G=PermutationGroup([(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,35),(2,36),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22),(25,37),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38)])`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D ··· 3H 4A 4B 6A 6B 6C 6D ··· 6H 6I 6J 6K 6L 6M 6N 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 2 2 2 2 3 3 3 3 ··· 3 4 4 6 6 6 6 ··· 6 6 6 6 6 6 6 12 12 12 ··· 12 12 12 size 1 1 9 9 18 18 18 18 2 2 2 4 ··· 4 2 18 2 2 2 4 ··· 4 18 18 36 36 36 36 2 2 4 ··· 4 18 18

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D12 S32 S3×D4 C2×S32 C32⋊4D6 S3×D12 D6⋊D6 C2×C32⋊4D6 C12⋊3S32 kernel C12⋊3S32 C33⋊9D4 C12×C3⋊S3 C3×C12⋊S3 C2×C32⋊4D6 C4×C3⋊S3 C12⋊S3 C3×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 1 2 2 1 2 2 1 3 5 4 3 2 3 2 4 2 2 4

Matrix representation of C123S32 in GL8(𝔽13)

 1 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 1 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 12 0 0 0 0 0 0 1 12
,
 1 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 10 10 0 0 0 0 0 0 12 3 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 0 0 11 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 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[10,12,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,11,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;`

C123S32 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_3S_3^2`
`% in TeX`

`G:=Group("C12:3S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,691);`
`// by ID`

`G=gap.SmallGroup(432,691);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,1124,571,2028,14118]);`
`// 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=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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