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## G = C12⋊2D12order 288 = 25·32

### 2nd semidirect product of C12 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊2D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C12⋊2D12
 Lower central C32 — C62 — C12⋊2D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C122D12
G = < a,b,c | a12=b12=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 946 in 215 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×10], C6 [×6], C6 [×5], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×6], C12 [×4], C12 [×4], D6 [×20], C2×C6 [×2], C2×C6 [×7], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4⋊Dic3, D6⋊C4 [×2], C6.D4 [×2], C3×C4⋊C4, S3×C2×C4 [×3], C2×D12, C2×D12 [×2], C2×C3⋊D4 [×2], C6×D4, C3⋊D12 [×4], C3×D12 [×2], C6×Dic3 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C12⋊D4, D63D4, D6⋊Dic3 [×2], C3×C4⋊Dic3, C2×C3⋊D12 [×2], C6×D12, C2×C4×C3⋊S3, C122D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, S3×D4 [×2], D42S3, Q83S3, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C12⋊D4, D63D4, D12⋊S3, D6⋊D6, C2×C3⋊D12, C122D12

Smallest permutation representation of C122D12
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 34 15 45 5 30 19 41 9 26 23 37)(2 33 16 44 6 29 20 40 10 25 24 48)(3 32 17 43 7 28 21 39 11 36 13 47)(4 31 18 42 8 27 22 38 12 35 14 46)
(1 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)(26 30)(27 35)(29 33)(32 36)(37 45)(39 43)(40 48)(42 46)```

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

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

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

42 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 ··· 12H 12I 12J 12K 12L 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 size 1 1 1 1 12 12 18 18 2 2 4 2 2 12 12 18 18 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C4○D4 D12 C3⋊D4 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C3⋊D12 C2×S32 D12⋊S3 D6⋊D6 kernel C12⋊2D12 D6⋊Dic3 C3×C4⋊Dic3 C2×C3⋊D12 C6×D12 C2×C4×C3⋊S3 C4⋊Dic3 C2×D12 C3×C12 C2×C3⋊S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C12 C2×C4 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 1 1 1 1 2 2 2 2 2 2 4 4 1 2 1 1 2 1 2 2

Matrix representation of C122D12 in GL8(𝔽13)

 8 3 0 0 0 0 0 0 0 5 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 12 0 0 0 0 0 0 0 0 12
,
 11 5 0 0 0 0 0 0 2 2 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 12 12 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 1 1
,
 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 12 1 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 12 0 0 0 0 0 0 0 1 1

`G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[11,2,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1] >;`

C122D12 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_2D_{12}`
`% in TeX`

`G:=Group("C12:2D12");`
`// GroupNames label`

`G:=SmallGroup(288,564);`
`// by ID`

`G=gap.SmallGroup(288,564);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,100,1356,9414]);`
`// Polycyclic`

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

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