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## G = C12.D12order 288 = 25·32

### 13rd non-split extension by C12 of 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 — C3×C12 — C6×C12 — C6×D12 — C12.D12
 Lower central C32 — C3×C6 — C62 — C12.D12
 Upper central C1 — C2 — C2×C4

Generators and relations for C12.D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >

Subgroups: 370 in 102 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], M4(2) [×2], C2×D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×5], C24, D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C3×C12 [×2], S3×C6 [×4], C62, C4.Dic3, C4.Dic3 [×3], C3×M4(2), C2×D12, C6×D4, C3×C3⋊C8, C324C8, C3×D12 [×2], C6×C12, S3×C2×C6 [×2], C12.46D4, C12.D4, C3×C4.Dic3, C12.58D6, C6×D12, C12.D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C4.D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C12.46D4, C12.D4, D6⋊Dic3, C12.D12

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B 12C ··· 12I 24A 24B 24C 24D order 1 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 2 12 12 2 2 4 2 2 2 2 2 2 4 4 4 4 12 12 12 12 12 12 36 36 2 2 4 ··· 4 12 12 12 12

39 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + - + - + image C1 C2 C2 C2 C4 S3 S3 D4 D6 Dic3 D12 C3⋊D4 C4×S3 C4.D4 S32 D6⋊S3 C3⋊D12 S3×Dic3 C12.46D4 C12.D4 C12.D12 kernel C12.D12 C3×C4.Dic3 C12.58D6 C6×D12 S3×C2×C6 C4.Dic3 C2×D12 C3×C12 C2×C12 C22×S3 C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 6 2 1 1 1 1 1 2 2 4

Matrix representation of C12.D12 in GL4(𝔽73) generated by

 24 0 0 0 0 70 0 0 0 15 49 0 15 0 0 3
,
 0 64 0 0 24 0 0 0 1 0 0 8 0 27 70 0
,
 1 0 0 57 0 27 6 0 0 29 46 0 20 0 0 72
`G:=sub<GL(4,GF(73))| [24,0,0,15,0,70,15,0,0,0,49,0,0,0,0,3],[0,24,1,0,64,0,0,27,0,0,0,70,0,0,8,0],[1,0,0,20,0,27,29,0,0,6,46,0,57,0,0,72] >;`

C12.D12 in GAP, Magma, Sage, TeX

`C_{12}.D_{12}`
`% in TeX`

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

`G:=SmallGroup(288,206);`
`// by ID`

`G=gap.SmallGroup(288,206);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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