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## G = D12.4D6order 288 = 25·32

### 4th non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.4D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.4D6
 Lower central C32 — C3×C6 — C3×C12 — D12.4D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for D12.4D6
G = < a,b,c,d | a12=b2=1, c6=a3, d2=a6, bab=a-1, ac=ca, dad-1=a7, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 530 in 130 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×4], C6 [×2], C6 [×2], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3 [×6], C12 [×2], C12 [×4], D6 [×4], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×3], Dic6 [×2], C4×S3 [×6], D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8 [×3], C8.C22, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3 [×3], C24⋊C2, Dic12, D4.S3, Q82S3, C3⋊Q16 [×2], C3×SD16, C3×Q16, D42S3, S3×Q8 [×2], Q83S3, C324C8, C3×C24, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C3×Dic6 [×3], C3×D12, C4×C3⋊S3, D4.D6, Q16⋊S3, Dic6⋊S3, C322Q16, C3×C24⋊C2, C3×Dic12, C24⋊S3, D12⋊S3, Dic3.D6, D12.4D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8.C22, S32, S3×D4 [×2], C2×S32, D4.D6, Q16⋊S3, D6⋊D6, D12.4D6

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C8.C22 S32 S3×D4 C2×S32 D4.D6 Q16⋊S3 D6⋊D6 D12.4D6 kernel D12.4D6 Dic6⋊S3 C32⋊2Q16 C3×C24⋊C2 C3×Dic12 C24⋊S3 D12⋊S3 Dic3.D6 C24⋊C2 Dic12 C3⋊Dic3 C2×C3⋊S3 C24 Dic6 D12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 2 1 2 2 2 4

Matrix representation of D12.4D6 in GL4(𝔽5) generated by

 0 0 2 0 1 2 0 0 4 4 4 1 4 4 0 4
,
 2 3 3 2 1 1 0 2 4 3 4 3 1 1 1 3
,
 4 4 0 1 4 3 1 0 4 1 4 2 2 4 4 0
,
 3 3 0 2 0 0 0 3 2 3 2 1 0 3 0 0
`G:=sub<GL(4,GF(5))| [0,1,4,4,0,2,4,4,2,0,4,0,0,0,1,4],[2,1,4,1,3,1,3,1,3,0,4,1,2,2,3,3],[4,4,4,2,4,3,1,4,0,1,4,4,1,0,2,0],[3,0,2,0,3,0,3,3,0,0,2,0,2,3,1,0] >;`

D12.4D6 in GAP, Magma, Sage, TeX

`D_{12}._4D_6`
`% in TeX`

`G:=Group("D12.4D6");`
`// GroupNames label`

`G:=SmallGroup(288,459);`
`// by ID`

`G=gap.SmallGroup(288,459);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,58,675,346,80,1356,9414]);`
`// Polycyclic`

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

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