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## G = D6.D12order 288 = 25·32

### 5th non-split extension by D6 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6.D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — D6.D12
 Lower central C32 — C62 — D6.D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D6.D12
G = < a,b,c,d | a6=b2=c12=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 762 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×6], C6 [×6], C6 [×5], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×6], C12 [×6], D6 [×2], D6 [×12], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×2], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C22.D4, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×3], C62, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4 [×5], C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3 [×2], C3⋊D12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.21D6, D6.D4, C6.D12, Dic3⋊Dic3, C3×C4⋊Dic3, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, C2×C3⋊D12, D6.D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C22×S3 [×2], C22.D4, S32, C2×D12, C4○D12, S3×D4, D42S3 [×2], Q83S3, C2×S32, C23.21D6, D6.D4, D12⋊S3, S3×D12, D6.3D6, D6.D12

Smallest permutation representation of D6.D12
On 48 points
Generators in S48
```(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 41 29 45 33 37)(26 42 30 46 34 38)(27 43 31 47 35 39)(28 44 32 48 36 40)
(1 27)(2 48)(3 29)(4 38)(5 31)(6 40)(7 33)(8 42)(9 35)(10 44)(11 25)(12 46)(13 47)(14 28)(15 37)(16 30)(17 39)(18 32)(19 41)(20 34)(21 43)(22 36)(23 45)(24 26)
(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 31 13 39)(2 30 14 38)(3 29 15 37)(4 28 16 48)(5 27 17 47)(6 26 18 46)(7 25 19 45)(8 36 20 44)(9 35 21 43)(10 34 22 42)(11 33 23 41)(12 32 24 40)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 12A ··· 12H 12I ··· 12N order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 6 6 36 2 2 4 4 6 6 12 12 18 18 2 ··· 2 4 4 4 12 12 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 D12 C4○D12 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 S3×D12 D6.3D6 kernel D6.D12 C6.D12 Dic3⋊Dic3 C3×C4⋊Dic3 C3×D6⋊C4 C6.11D12 C2×S3×Dic3 C2×C3⋊D12 C4⋊Dic3 D6⋊C4 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 2 1 4 4 4 1 1 2 1 1 2 2 2

Matrix representation of D6.D12 in GL8(𝔽13)

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

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

D6.D12 in GAP, Magma, Sage, TeX

`D_6.D_{12}`
`% in TeX`

`G:=Group("D6.D12");`
`// GroupNames label`

`G:=SmallGroup(288,538);`
`// by ID`

`G=gap.SmallGroup(288,538);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,1356,9414]);`
`// Polycyclic`

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

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