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## G = D12⋊6D6order 288 = 25·32

### 6th semidirect product of D12 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊6D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — D12⋊6D6
 Lower central C32 — C3×C6 — C3×C12 — D12⋊6D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D126D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c-1 >

Subgroups: 714 in 146 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×6], C6 [×2], C6 [×4], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3, C12 [×2], C12 [×5], D6, D6 [×9], C2×C6 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3⋊S3, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, C4×S3, C4×S3, D12 [×2], D12 [×5], C3⋊D4, C2×C12 [×2], C3×D4 [×3], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3×C12, C3×C12, S32 [×2], S3×C6, S3×C6 [×2], C2×C3⋊S3, C8⋊S3, D24, C4.Dic3, D4⋊S3 [×3], Q82S3, Q82S3 [×3], C3×SD16, C2×D12, S3×D4, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, C3⋊D12, S3×C12, S3×C12, C3×D12 [×2], C3×D12, C12⋊S3, Q8×C32, C2×S32, Q83D6, D4⋊D6, D6.Dic3, C322D8, C3⋊D24, C3×Q82S3, C3211SD16, S3×D12, C3×Q83S3, D126D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q83D6, D4⋊D6, S3×C3⋊D4, D126D6

Smallest permutation representation of D126D6
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 35 5 31 9 27)(2 28 6 36 10 32)(3 33 7 29 11 25)(4 26 8 34 12 30)(13 38 17 46 21 42)(14 43 18 39 22 47)(15 48 19 44 23 40)(16 41 20 37 24 45)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 33)(26 32)(27 31)(28 30)(34 36)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 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 C3⋊D4 C3⋊D4 C8⋊C22 S32 S3×D4 C2×S32 Q8⋊3D6 D4⋊D6 S3×C3⋊D4 D12⋊6D6 kernel D12⋊6D6 D6.Dic3 C32⋊2D8 C3⋊D24 C3×Q8⋊2S3 C32⋊11SD16 S3×D12 C3×Q8⋊3S3 Q8⋊2S3 Q8⋊3S3 C3×Dic3 S3×C6 C3⋊C8 C4×S3 D12 C3×Q8 Dic3 D6 C32 Q8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of D126D6 in GL8(ℤ)

 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0
,
 0 0 0 0 1 0 0 1 0 0 0 0 1 -1 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 1 -1 0 1 -1 0 0 0 0 0 1 0 -1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 -1 1 0 0 0 0 0
,
 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 0
,
 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 1

`G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,-1,0,0,0,0,-1,-1,0,1,0,0,0,0,0,1,1,0,1,1,0,-1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,-1,0,0,0,0],[0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0],[-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1] >;`

D126D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_6D_6`
`% in TeX`

`G:=Group("D12:6D6");`
`// GroupNames label`

`G:=SmallGroup(288,587);`
`// by ID`

`G=gap.SmallGroup(288,587);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^-1>;`
`// generators/relations`

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