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## G = D6.1D12order 288 = 25·32

### 1st non-split extension by D6 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D6.1D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D6.1D12
 Lower central C32 — C3×C6 — C3×C12 — D6.1D12
 Upper central C1 — C2 — C4 — C8

Generators and relations for D6.1D12
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=c11 >

Subgroups: 618 in 134 conjugacy classes, 40 normal (all 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, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C24⋊C2, D24, Dic12, D4⋊S3, C3⋊Q16, C2×C24, C3×SD16, C4○D12, D42S3, Q83S3, C3×C3⋊C8, C3×C24, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C324Q8, C12⋊S3, C4○D24, Q8.7D6, C3⋊D24, C323Q16, S3×C24, C3×C24⋊C2, C242S3, D125S3, D6.6D6, D6.1D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, Q8.7D6, S3×D12, D6.1D12

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 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 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 D12 D12 C4○D8 C4○D24 S32 S3×D4 C2×S32 Q8.7D6 S3×D12 D6.1D12 kernel D6.1D12 C3⋊D24 C32⋊3Q16 S3×C24 C3×C24⋊C2 C24⋊2S3 D12⋊5S3 D6.6D6 S3×C8 C24⋊C2 C3×Dic3 S3×C6 C3⋊C8 C24 Dic6 C4×S3 D12 Dic3 D6 C32 C3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 4 8 1 1 1 2 2 4

Matrix representation of D6.1D12 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
,
 56 8 0 0 0 0 37 17 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 55 0 0 0 0 8 59 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 46 0 0 0 0 0 13 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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],[56,37,0,0,0,0,8,17,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,8,0,0,0,0,55,59,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,13,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;`

D6.1D12 in GAP, Magma, Sage, TeX

`D_6._1D_{12}`
`% in TeX`

`G:=Group("D6.1D12");`
`// GroupNames label`

`G:=SmallGroup(288,454);`
`// by ID`

`G=gap.SmallGroup(288,454);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,58,346,80,1356,9414]);`
`// 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=c^11>;`
`// generators/relations`

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