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## G = C12⋊S32order 432 = 24·33

### 2nd semidirect product of C12 and S32 acting via S32/C32=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12⋊S32
G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, cac=a-1, ad=da, eae=a7, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2624 in 388 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C3 [×4], C4, C4, C22 [×9], S3 [×28], C6, C6 [×4], C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C32 [×4], Dic3 [×9], C12, C12 [×4], C12 [×4], D6 [×2], D6 [×37], C2×C6 [×10], C2×D4, C3×S3 [×16], C3⋊S3 [×20], C3×C6, C3×C6 [×4], C3×C6 [×6], C4×S3 [×9], D12, D12 [×4], C3⋊D4 [×10], C3×D4 [×5], C22×S3 [×10], C33, C3⋊Dic3 [×9], C3×C12, C3×C12 [×4], C3×C12 [×4], S32 [×16], S3×C6 [×8], S3×C6 [×8], C2×C3⋊S3 [×2], C2×C3⋊S3 [×13], C62 [×2], S3×D4 [×5], S3×C32 [×2], C3×C3⋊S3 [×2], C33⋊C2 [×2], C32×C6, D6⋊S3 [×8], C3×D12 [×4], C3×D12 [×4], C4×C3⋊S3 [×9], C12⋊S3, C327D4 [×2], D4×C32, C2×S32 [×8], C22×C3⋊S3 [×2], C335C4, C32×C12, S3×C3⋊S3 [×4], S3×C3×C6 [×2], C6×C3⋊S3 [×2], C2×C33⋊C2, D6⋊D6 [×4], D4×C3⋊S3, C336D4 [×2], C32×D12, C3×C12⋊S3, C4×C33⋊C2, C2×S3×C3⋊S3 [×2], C12⋊S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], D4 [×2], C23, D6 [×15], C2×D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], S3×D4 [×5], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D6⋊D6 [×4], D4×C3⋊S3, C2×S3×C3⋊S3, C12⋊S32

Smallest permutation representation of C12⋊S32
On 72 points
Generators in S72
```(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 20 63)(2 21 64)(3 22 65)(4 23 66)(5 24 67)(6 13 68)(7 14 69)(8 15 70)(9 16 71)(10 17 72)(11 18 61)(12 19 62)(25 48 55)(26 37 56)(27 38 57)(28 39 58)(29 40 59)(30 41 60)(31 42 49)(32 43 50)(33 44 51)(34 45 52)(35 46 53)(36 47 54)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 60)(9 59)(10 58)(11 57)(12 56)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)(25 63)(26 62)(27 61)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)
(1 24 71)(2 13 72)(3 14 61)(4 15 62)(5 16 63)(6 17 64)(7 18 65)(8 19 66)(9 20 67)(10 21 68)(11 22 69)(12 23 70)(25 51 40)(26 52 41)(27 53 42)(28 54 43)(29 55 44)(30 56 45)(31 57 46)(32 58 47)(33 59 48)(34 60 37)(35 49 38)(36 50 39)
(1 55)(2 50)(3 57)(4 52)(5 59)(6 54)(7 49)(8 56)(9 51)(10 58)(11 53)(12 60)(13 36)(14 31)(15 26)(16 33)(17 28)(18 35)(19 30)(20 25)(21 32)(22 27)(23 34)(24 29)(37 70)(38 65)(39 72)(40 67)(41 62)(42 69)(43 64)(44 71)(45 66)(46 61)(47 68)(48 63)```

`G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,20,63)(2,21,64)(3,22,65)(4,23,66)(5,24,67)(6,13,68)(7,14,69)(8,15,70)(9,16,71)(10,17,72)(11,18,61)(12,19,62)(25,48,55)(26,37,56)(27,38,57)(28,39,58)(29,40,59)(30,41,60)(31,42,49)(32,43,50)(33,44,51)(34,45,52)(35,46,53)(36,47,54), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,60)(9,59)(10,58)(11,57)(12,56)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,63)(26,62)(27,61)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64), (1,24,71)(2,13,72)(3,14,61)(4,15,62)(5,16,63)(6,17,64)(7,18,65)(8,19,66)(9,20,67)(10,21,68)(11,22,69)(12,23,70)(25,51,40)(26,52,41)(27,53,42)(28,54,43)(29,55,44)(30,56,45)(31,57,46)(32,58,47)(33,59,48)(34,60,37)(35,49,38)(36,50,39), (1,55)(2,50)(3,57)(4,52)(5,59)(6,54)(7,49)(8,56)(9,51)(10,58)(11,53)(12,60)(13,36)(14,31)(15,26)(16,33)(17,28)(18,35)(19,30)(20,25)(21,32)(22,27)(23,34)(24,29)(37,70)(38,65)(39,72)(40,67)(41,62)(42,69)(43,64)(44,71)(45,66)(46,61)(47,68)(48,63)>;`

`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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,20,63)(2,21,64)(3,22,65)(4,23,66)(5,24,67)(6,13,68)(7,14,69)(8,15,70)(9,16,71)(10,17,72)(11,18,61)(12,19,62)(25,48,55)(26,37,56)(27,38,57)(28,39,58)(29,40,59)(30,41,60)(31,42,49)(32,43,50)(33,44,51)(34,45,52)(35,46,53)(36,47,54), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,60)(9,59)(10,58)(11,57)(12,56)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,63)(26,62)(27,61)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64), (1,24,71)(2,13,72)(3,14,61)(4,15,62)(5,16,63)(6,17,64)(7,18,65)(8,19,66)(9,20,67)(10,21,68)(11,22,69)(12,23,70)(25,51,40)(26,52,41)(27,53,42)(28,54,43)(29,55,44)(30,56,45)(31,57,46)(32,58,47)(33,59,48)(34,60,37)(35,49,38)(36,50,39), (1,55)(2,50)(3,57)(4,52)(5,59)(6,54)(7,49)(8,56)(9,51)(10,58)(11,53)(12,60)(13,36)(14,31)(15,26)(16,33)(17,28)(18,35)(19,30)(20,25)(21,32)(22,27)(23,34)(24,29)(37,70)(38,65)(39,72)(40,67)(41,62)(42,69)(43,64)(44,71)(45,66)(46,61)(47,68)(48,63) );`

`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),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,20,63),(2,21,64),(3,22,65),(4,23,66),(5,24,67),(6,13,68),(7,14,69),(8,15,70),(9,16,71),(10,17,72),(11,18,61),(12,19,62),(25,48,55),(26,37,56),(27,38,57),(28,39,58),(29,40,59),(30,41,60),(31,42,49),(32,43,50),(33,44,51),(34,45,52),(35,46,53),(36,47,54)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,60),(9,59),(10,58),(11,57),(12,56),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,48),(21,47),(22,46),(23,45),(24,44),(25,63),(26,62),(27,61),(28,72),(29,71),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64)], [(1,24,71),(2,13,72),(3,14,61),(4,15,62),(5,16,63),(6,17,64),(7,18,65),(8,19,66),(9,20,67),(10,21,68),(11,22,69),(12,23,70),(25,51,40),(26,52,41),(27,53,42),(28,54,43),(29,55,44),(30,56,45),(31,57,46),(32,58,47),(33,59,48),(34,60,37),(35,49,38),(36,50,39)], [(1,55),(2,50),(3,57),(4,52),(5,59),(6,54),(7,49),(8,56),(9,51),(10,58),(11,53),(12,60),(13,36),(14,31),(15,26),(16,33),(17,28),(18,35),(19,30),(20,25),(21,32),(22,27),(23,34),(24,29),(37,70),(38,65),(39,72),(40,67),(41,62),(42,69),(43,64),(44,71),(45,66),(46,61),(47,68),(48,63)])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S 12A ··· 12M order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 12 ··· 12 size 1 1 6 6 18 18 27 27 2 ··· 2 4 4 4 4 2 54 2 ··· 2 4 4 4 4 12 ··· 12 36 36 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 S32 S3×D4 C2×S32 D6⋊D6 kernel C12⋊S32 C33⋊6D4 C32×D12 C3×C12⋊S3 C4×C33⋊C2 C2×S3×C3⋊S3 C3×D12 C12⋊S3 C33⋊C2 C3×C12 S3×C6 C2×C3⋊S3 C12 C32 C6 C3 # reps 1 2 1 1 1 2 4 1 2 5 8 2 4 5 4 8

Matrix representation of C12⋊S32 in GL8(𝔽13)

 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 3 0 0 0 0 0 0 8 1 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 0 1
,
 12 1 0 0 0 0 0 0 12 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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 1 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

`G:=sub<GL(8,GF(13))| [0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;`

C12⋊S32 in GAP, Magma, Sage, TeX

`C_{12}\rtimes S_3^2`
`% in TeX`

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

`G:=SmallGroup(432,673);`
`// by ID`

`G=gap.SmallGroup(432,673);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e=a^7,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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