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## G = D12⋊12D6order 288 = 25·32

### 6th semidirect product of D12 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊12D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×C3⋊D4 — D12⋊12D6
 Lower central C32 — C3×C6 — D12⋊12D6
 Upper central C1 — C2 — D4

Generators and relations for D1212D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 1338 in 352 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, S3×D4, S3×D4, D42S3, C2×C3⋊D4, C6×D4, S3×Dic3, D6⋊S3, D6⋊S3, C3⋊D12, C322Q8, S3×C12, C3×D12, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C2×S32, S3×C2×C6, D46D6, D125S3, D6.D6, D6⋊D6, D6.4D6, C2×D6⋊S3, S3×C3⋊D4, C3×S3×D4, C12.D6, D1212D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D46D6, C22×S32, D1212D6

Smallest permutation representation of D1212D6
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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)
(1 27 5 31 9 35)(2 34 6 26 10 30)(3 29 7 33 11 25)(4 36 8 28 12 32)(13 48 21 44 17 40)(14 43 22 39 18 47)(15 38 23 46 19 42)(16 45 24 41 20 37)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 12A 12B 12C 12D 12E order 1 2 2 2 2 ··· 2 2 3 3 3 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 size 1 1 2 2 6 ··· 6 18 2 2 4 2 6 6 18 18 18 2 2 4 ··· 4 6 6 6 6 8 8 12 12 12 12 4 4 8 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 2+ 1+4 S32 C2×S32 C2×S32 D4⋊6D6 D12⋊12D6 kernel D12⋊12D6 D12⋊5S3 D6.D6 D6⋊D6 D6.4D6 C2×D6⋊S3 S3×C3⋊D4 C3×S3×D4 C12.D6 S3×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 C32 D4 C4 C22 C3 C1 # reps 1 2 1 1 2 2 4 2 1 2 2 2 4 2 4 1 1 1 2 4 1

Matrix representation of D1212D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 11 12 6 0 0 2 11 7 6 0 0 6 3 4 2 0 0 10 3 11 2
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 1
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 8 0 1 0 0 0 0 8 0 1
,
 1 0 0 0 0 0 1 12 0 0 0 0 0 0 11 9 0 0 0 0 4 2 0 0 0 0 0 0 11 9 0 0 0 0 4 2

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,6,10,0,0,11,11,3,3,0,0,12,7,4,11,0,0,6,6,2,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,8,0,0,0,0,12,0,8,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;`

D1212D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{12}D_6`
`% in TeX`

`G:=Group("D12:12D6");`
`// GroupNames label`

`G:=SmallGroup(288,961);`
`// by ID`

`G=gap.SmallGroup(288,961);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,1356,9414]);`
`// Polycyclic`

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

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