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## G = D12⋊9D6order 288 = 25·32

### 3rd semidirect product of D12 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊9D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12⋊9D6
 Lower central C32 — C3×C6 — C3×C12 — D12⋊9D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D129D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >

Subgroups: 586 in 146 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6 [×2], C6 [×7], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3, Dic3 [×3], C12 [×2], C12 [×2], D6, D6 [×4], C2×C6 [×8], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6 [×3], C4×S3, C4×S3, D12 [×2], C2×Dic3, C3⋊D4 [×3], C2×C12, C3×D4 [×2], C3×D4 [×4], C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6 [×4], C62, C8⋊S3, C24⋊C2, C4.Dic3, D4⋊S3, D4⋊S3 [×2], D4.S3 [×4], C3×D8, C4○D12, S3×D4, D42S3, C6×D4, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, S3×C12, C3×D12 [×2], C3×C3⋊D4, C324Q8, D4×C32, S3×C2×C6, D8⋊S3, D126C22, D6.Dic3, C322D8, D12.S3, C3×D4⋊S3, C329SD16, D125S3, C3×S3×D4, D129D6
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, D8⋊S3, D126C22, S3×C3⋊D4, D129D6

Smallest permutation representation of D129D6
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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(25 37)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 14 21 22 17 18)(15 16 23 24 19 20)(25 29 33)(26 36 34 32 30 28)(27 31 35)(37 42 41 46 45 38)(39 44 43 48 47 40)
(1 31)(2 26)(3 33)(4 28)(5 35)(6 30)(7 25)(8 32)(9 27)(10 34)(11 29)(12 36)(13 44)(14 39)(15 46)(16 41)(17 48)(18 43)(19 38)(20 45)(21 40)(22 47)(23 42)(24 37)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 12A 12B 12C 12D 24A 24B order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 12 24 24 size 1 1 4 6 12 12 2 2 4 2 6 36 2 2 4 4 4 6 6 8 8 8 12 12 24 12 36 4 4 8 12 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 D8⋊S3 D12⋊6C22 S3×C3⋊D4 D12⋊9D6 kernel D12⋊9D6 D6.Dic3 C32⋊2D8 D12.S3 C3×D4⋊S3 C32⋊9SD16 D12⋊5S3 C3×S3×D4 D4⋊S3 S3×D4 C3×Dic3 S3×C6 C3⋊C8 C4×S3 D12 C3×D4 Dic3 D6 C32 D4 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 D129D6 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 72 1 2 0 0 0 0 0 1 72 72
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 72 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 1 1
,
 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 72 1 2 0 0 0 0 1 0 0 72
,
 1 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 37 42 62 62 0 0 0 0 31 36 0 11 0 0 0 0 37 37 5 10 0 0 0 0 0 36 31 68

`G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,1,72,0,0,0,0,1,1,0,1,0,0,0,0,2,0,0,1],[1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,72,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,72],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,37,31,37,0,0,0,0,0,42,36,37,36,0,0,0,0,62,0,5,31,0,0,0,0,62,11,10,68] >;`

D129D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_9D_6`
`% in TeX`

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

`G:=SmallGroup(288,580);`
`// by ID`

`G=gap.SmallGroup(288,580);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,346,185,80,1356,9414]);`
`// 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=d*a*d=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;`
`// generators/relations`

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