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## G = D12.14D6order 288 = 25·32

### 14th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.14D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.14D6
 Lower central C32 — C3×C6 — C3×C12 — D12.14D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.14D6
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 666 in 144 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×8], C6 [×2], C6 [×2], C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, C32, Dic3 [×4], C12 [×2], C12 [×6], D6 [×8], C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×8], D12, D12 [×8], C2×Dic3, C3⋊D4, C3×D4, C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, S3×C8 [×2], C24⋊C2, D24, D4⋊S3, Q82S3, Q82S3, C3⋊Q16, C3×SD16, C3×Q16, D42S3, Q83S3 [×4], C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, Q8.7D6, D24⋊C2, C12.29D6, C3⋊D24, C325SD16, C3×Q82S3, C3×C3⋊Q16, D12⋊S3, C12.26D6, D12.14D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, Q8.7D6, D24⋊C2, Dic3⋊D6, D12.14D6

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

33 conjugacy classes

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

33 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 8 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 C4○D8 S32 S3×D4 C2×S32 Q8.7D6 D24⋊C2 Dic3⋊D6 D12.14D6 kernel D12.14D6 C12.29D6 C3⋊D24 C32⋊5SD16 C3×Q8⋊2S3 C3×C3⋊Q16 D12⋊S3 C12.26D6 Q8⋊2S3 C3⋊Q16 C3⋊Dic3 C2×C3⋊S3 C3⋊C8 Dic6 D12 C3×Q8 C32 Q8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 4 1 2 1 2 2 2 1

Matrix representation of D12.14D6 in GL6(𝔽73)

 72 3 0 0 0 0 48 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 25 0 0 0 0 38 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 27 0 0 0 0 0 18 46 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 1
,
 41 48 0 0 0 0 38 0 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))| [72,48,0,0,0,0,3,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,38,0,0,0,0,25,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,18,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[41,38,0,0,0,0,48,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D12.14D6 in GAP, Magma, Sage, TeX

`D_{12}._{14}D_6`
`% in TeX`

`G:=Group("D12.14D6");`
`// GroupNames label`

`G:=SmallGroup(288,598);`
`// by ID`

`G=gap.SmallGroup(288,598);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,100,675,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^3,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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