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## G = C12⋊D8order 192 = 26·3

### 3rd semidirect product of C12 and D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12⋊D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C12⋊D8
 Lower central C3 — C6 — C2×C12 — C12⋊D8
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C12⋊D8
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 592 in 162 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C22, C22 [×12], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], C12 [×6], D6 [×6], C2×C6, C2×C6 [×6], C42, C2×C8 [×2], D8 [×8], C2×D4 [×2], C2×D4 [×6], C3⋊C8 [×4], D12 [×8], C2×C12, C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×C6 [×2], C4×C8, C41D4, C41D4, C2×D8 [×4], C2×C3⋊C8 [×2], D4⋊S3 [×8], C4×C12, C2×D12 [×2], C2×D12 [×2], C6×D4 [×2], C6×D4 [×2], C84D4, C4×C3⋊C8, C4⋊D12, C2×D4⋊S3 [×4], C3×C41D4, C12⋊D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], D8 [×4], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C2×D8 [×2], D4⋊S3 [×4], S3×D4 [×2], C2×C3⋊D4, C84D4, C2×D4⋊S3 [×2], C123D4, C12⋊D8

Smallest permutation representation of C12⋊D8
On 96 points
Generators in S96
```(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 15 73 26 90 67 49 39)(2 20 74 31 91 72 50 44)(3 13 75 36 92 65 51 37)(4 18 76 29 93 70 52 42)(5 23 77 34 94 63 53 47)(6 16 78 27 95 68 54 40)(7 21 79 32 96 61 55 45)(8 14 80 25 85 66 56 38)(9 19 81 30 86 71 57 43)(10 24 82 35 87 64 58 48)(11 17 83 28 88 69 59 41)(12 22 84 33 89 62 60 46)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 41)(14 40)(15 39)(16 38)(17 37)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 72)(34 71)(35 70)(36 69)(49 73)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(85 95)(86 94)(87 93)(88 92)(89 91)```

`G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,15,73,26,90,67,49,39)(2,20,74,31,91,72,50,44)(3,13,75,36,92,65,51,37)(4,18,76,29,93,70,52,42)(5,23,77,34,94,63,53,47)(6,16,78,27,95,68,54,40)(7,21,79,32,96,61,55,45)(8,14,80,25,85,66,56,38)(9,19,81,30,86,71,57,43)(10,24,82,35,87,64,58,48)(11,17,83,28,88,69,59,41)(12,22,84,33,89,62,60,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,72)(34,71)(35,70)(36,69)(49,73)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(85,95)(86,94)(87,93)(88,92)(89,91)>;`

`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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,15,73,26,90,67,49,39)(2,20,74,31,91,72,50,44)(3,13,75,36,92,65,51,37)(4,18,76,29,93,70,52,42)(5,23,77,34,94,63,53,47)(6,16,78,27,95,68,54,40)(7,21,79,32,96,61,55,45)(8,14,80,25,85,66,56,38)(9,19,81,30,86,71,57,43)(10,24,82,35,87,64,58,48)(11,17,83,28,88,69,59,41)(12,22,84,33,89,62,60,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,72)(34,71)(35,70)(36,69)(49,73)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(85,95)(86,94)(87,93)(88,92)(89,91) );`

`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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,15,73,26,90,67,49,39),(2,20,74,31,91,72,50,44),(3,13,75,36,92,65,51,37),(4,18,76,29,93,70,52,42),(5,23,77,34,94,63,53,47),(6,16,78,27,95,68,54,40),(7,21,79,32,96,61,55,45),(8,14,80,25,85,66,56,38),(9,19,81,30,86,71,57,43),(10,24,82,35,87,64,58,48),(11,17,83,28,88,69,59,41),(12,22,84,33,89,62,60,46)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,41),(14,40),(15,39),(16,38),(17,37),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,72),(34,71),(35,70),(36,69),(49,73),(50,84),(51,83),(52,82),(53,81),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74),(85,95),(86,94),(87,93),(88,92),(89,91)])`

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4F 6A 6B 6C 6D 6E 6F 6G 8A ··· 8H 12A ··· 12F order 1 2 2 2 2 2 2 2 3 4 ··· 4 6 6 6 6 6 6 6 8 ··· 8 12 ··· 12 size 1 1 1 1 8 8 24 24 2 2 ··· 2 2 2 2 8 8 8 8 6 ··· 6 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 C3⋊D4 D4⋊S3 S3×D4 kernel C12⋊D8 C4×C3⋊C8 C4⋊D12 C2×D4⋊S3 C3×C4⋊1D4 C4⋊1D4 C3⋊C8 C2×C12 C42 C2×D4 C12 C2×C4 C4 C4 # reps 1 1 1 4 1 1 4 2 1 2 8 4 4 2

Matrix representation of C12⋊D8 in GL6(𝔽73)

 0 72 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 30 60 0 0 0 0 30 43 0 0 0 0 0 0 16 57 0 0 0 0 16 16 0 0 0 0 0 0 57 16 0 0 0 0 57 57
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,30,0,0,0,0,60,43,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,57,0,0,0,0,16,57],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;`

C12⋊D8 in GAP, Magma, Sage, TeX

`C_{12}\rtimes D_8`
`% in TeX`

`G:=Group("C12:D8");`
`// GroupNames label`

`G:=SmallGroup(192,632);`
`// by ID`

`G=gap.SmallGroup(192,632);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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