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## G = D6.3D12order 288 = 25·32

### 3rd non-split extension by D6 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D6.3D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — D6.3D12
 Lower central C32 — C3×C6 — C3×C12 — D6.3D12
 Upper central C1 — C2 — C4 — C8

Generators and relations for D6.3D12
G = < a,b,c,d | a6=b2=1, c12=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c11 >

Subgroups: 674 in 135 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, D24, Dic12, Q82S3, C2×C24, C3×Q16, C4○D12, Q83S3, C3×C3⋊C8, C3×C24, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, C4○D24, D24⋊C2, C325SD16, S3×C24, C3×Dic12, C325D8, D6.6D6, D6.3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, D24⋊C2, S3×D12, D6.3D12

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D12 D12 C4○D8 C4○D24 S32 S3×D4 C2×S32 D24⋊C2 S3×D12 D6.3D12 kernel D6.3D12 C32⋊5SD16 S3×C24 C3×Dic12 C32⋊5D8 D6.6D6 S3×C8 Dic12 C3×Dic3 S3×C6 C3⋊C8 C24 Dic6 C4×S3 Dic3 D6 C32 C3 C8 C6 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 1 1 1 1 2 2 1 2 2 4 8 1 1 1 2 2 4

Matrix representation of D6.3D12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 27 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 57 16 0 0 0 0 57 57 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 67 67 0 0 0 0 67 6 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72 1

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[67,67,0,0,0,0,67,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;`

D6.3D12 in GAP, Magma, Sage, TeX

`D_6._3D_{12}`
`% in TeX`

`G:=Group("D6.3D12");`
`// GroupNames label`

`G:=SmallGroup(288,456);`
`// by ID`

`G=gap.SmallGroup(288,456);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,142,346,80,1356,9414]);`
`// Polycyclic`

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

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