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## G = D12.9D6order 288 = 25·32

### 9th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.9D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.9D6
 Lower central C32 — C3×C6 — C3×C12 — D12.9D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.9D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 738 in 155 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×8], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8 [×2], C23, C32, Dic3 [×7], C12 [×2], C12 [×5], D6 [×9], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×7], C4×S3 [×7], D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C2×SD16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8 [×2], C24⋊C2 [×2], D4.S3 [×2], Q82S3 [×2], C3×SD16 [×2], S3×D4 [×2], S3×Q8 [×3], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×S32, S3×SD16 [×2], C12.29D6, D12.S3 [×2], C3×Q82S3 [×2], D6⋊D6, Q8×C3⋊S3, D12.9D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C22×S3 [×2], C2×SD16, S32, S3×D4 [×2], C2×S32, S3×SD16 [×2], Dic3⋊D6, D12.9D6

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C ··· 12G 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 9 9 12 12 2 2 4 2 4 18 36 2 2 4 24 24 6 6 6 6 4 4 8 ··· 8 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 S32 S3×D4 C2×S32 S3×SD16 Dic3⋊D6 D12.9D6 kernel D12.9D6 C12.29D6 D12.S3 C3×Q8⋊2S3 D6⋊D6 Q8×C3⋊S3 Q8⋊2S3 C3⋊Dic3 C2×C3⋊S3 C3⋊C8 D12 C3×Q8 C3⋊S3 Q8 C6 C4 C3 C2 C1 # reps 1 1 2 2 1 1 2 1 1 2 2 2 4 1 2 1 4 2 1

Matrix representation of D12.9D6 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 3 0 0 0 0 48 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 72 1 0 0 0 0 0 0 62 68 0 0 0 0 24 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 21 0 0 0 0 33 67 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 55 0 0 0 0 4 0 0 0 0 0 0 0 72 0 0 0 0 0 72 1

`G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,62,24,0,0,0,0,68,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,33,0,0,0,0,21,67,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,55,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;`

D12.9D6 in GAP, Magma, Sage, TeX

`D_{12}._9D_6`
`% in TeX`

`G:=Group("D12.9D6");`
`// GroupNames label`

`G:=SmallGroup(288,588);`
`// by ID`

`G=gap.SmallGroup(288,588);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,100,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;`
`// generators/relations`

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