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## G = C12⋊D12order 288 = 25·32

### 1st semidirect product of C12 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C12⋊D12
 Lower central C32 — C62 — C12⋊D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C12⋊D12
G = < a,b,c | a12=b12=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 1250 in 243 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×10], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×2], D4 [×12], C23 [×4], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×26], C2×C6 [×2], C2×C6 [×7], C42, C2×D4 [×6], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C41D4, C3×Dic3 [×4], C3×C12 [×2], S3×C6 [×6], C2×C3⋊S3 [×6], C62, C4×Dic3, C4×C12, C2×D12, C2×D12 [×7], C2×C3⋊D4 [×4], C6×D4, C3⋊D12 [×8], C3×D12 [×2], C6×Dic3 [×2], C12⋊S3 [×2], C6×C12, S3×C2×C6 [×2], C22×C3⋊S3 [×2], C4⋊D12, C123D4, Dic3×C12, C2×C3⋊D12 [×4], C6×D12, C2×C12⋊S3, C12⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×6], C3⋊D4 [×2], C22×S3 [×2], C41D4, S32, C2×D12 [×3], S3×D4 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4⋊D12, C123D4, S3×D12 [×2], C2×C3⋊D12, C12⋊D12

Smallest permutation representation of C12⋊D12
On 48 points
Generators in S48
```(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 28 47 15 5 36 39 23 9 32 43 19)(2 33 48 20 6 29 40 16 10 25 44 24)(3 26 37 13 7 34 41 21 11 30 45 17)(4 31 38 18 8 27 42 14 12 35 46 22)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 21)(14 20)(15 19)(16 18)(22 24)(25 31)(26 30)(27 29)(32 36)(33 35)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D12 D12 C3⋊D4 S32 S3×D4 C3⋊D12 C2×S32 S3×D12 kernel C12⋊D12 Dic3×C12 C2×C3⋊D12 C6×D12 C2×C12⋊S3 C4×Dic3 C2×D12 C3×Dic3 C3×C12 C2×Dic3 C2×C12 C22×S3 Dic3 C12 C12 C2×C4 C6 C4 C22 C2 # reps 1 1 4 1 1 1 1 4 2 2 2 2 8 4 4 1 2 2 1 4

Matrix representation of C12⋊D12 in GL8(𝔽13)

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

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

C12⋊D12 in GAP, Magma, Sage, TeX

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

`G:=Group("C12:D12");`
`// GroupNames label`

`G:=SmallGroup(288,559);`
`// by ID`

`G=gap.SmallGroup(288,559);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,100,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=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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