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## G = D12.27D6order 288 = 25·32

### 2nd non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.27D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — D12.27D6
 Lower central C32 — C3×C6 — C3×C12 — D12.27D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D12.27D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >

Subgroups: 602 in 144 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×5], C6 [×2], C6 [×5], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C32, Dic3 [×5], C12 [×4], C12 [×3], D6 [×5], C2×C6 [×2], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×3], C4×S3 [×5], D12, D12 [×3], C3⋊D4 [×5], C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12 [×2], S3×C6, C2×C3⋊S3, C62, C24⋊C2 [×2], D24, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C4○D12, C4○D12 [×3], C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C4○D24, Q8.13D6, C3⋊D24, D12.S3, C325SD16, C323Q16, C6×C3⋊C8, C3×C4○D12, C12.59D6, D12.27D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4○D8, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4○D24, Q8.13D6, C2×C3⋊D12, D12.27D6

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 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 D12 C3⋊D4 D12 C3⋊D4 C4○D8 C4○D24 S32 C3⋊D12 C2×S32 C3⋊D12 Q8.13D6 D12.27D6 kernel D12.27D6 C3⋊D24 D12.S3 C32⋊5SD16 C32⋊3Q16 C6×C3⋊C8 C3×C4○D12 C12.59D6 C2×C3⋊C8 C4○D12 C3×C12 C62 C3⋊C8 Dic6 D12 C2×C12 C12 C12 C2×C6 C2×C6 C32 C3 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 4 8 1 1 1 1 2 4

Matrix representation of D12.27D6 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 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
,
 16 16 0 0 0 0 16 57 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
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

D12.27D6 in GAP, Magma, Sage, TeX

`D_{12}._{27}D_6`
`% in TeX`

`G:=Group("D12.27D6");`
`// GroupNames label`

`G:=SmallGroup(288,477);`
`// by ID`

`G=gap.SmallGroup(288,477);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,100,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,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^5>;`
`// generators/relations`

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