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## G = D12⋊24D6order 288 = 25·32

### 8th semidirect product of D12 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊24D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×C3⋊D4 — D12⋊24D6
 Lower central C32 — C3×C6 — D12⋊24D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D1224D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=cac-1=a-1, dad=a5, cbc-1=a4b, dbd=a10b, dcd=c-1 >

Subgroups: 1434 in 352 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×12], C6 [×2], C6 [×10], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], C32, Dic3 [×2], Dic3 [×6], C12 [×4], C12 [×4], D6 [×6], D6 [×16], C2×C6 [×2], C2×C6 [×9], C2×D4 [×9], C4○D4 [×6], C3×S3 [×6], C3⋊S3 [×2], C3×C6, C3×C6, Dic6, Dic6 [×3], C4×S3 [×2], C4×S3 [×10], D12, D12 [×4], D12 [×7], C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×18], C2×C12 [×2], C2×C12 [×3], C3×D4 [×7], C3×Q8, C22×S3 [×2], C22×S3 [×8], C22×C6 [×2], 2+ 1+4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S32 [×4], S3×C6 [×6], S3×C6 [×2], C2×C3⋊S3 [×2], C62, C2×D12, C2×D12 [×2], C4○D12, C4○D12 [×5], S3×D4 [×10], D42S3 [×4], Q83S3 [×2], C2×C3⋊D4 [×4], C6×D4, C3×C4○D4, S3×Dic3 [×4], D6⋊S3 [×4], C3⋊D12 [×4], C3×Dic6, S3×C12 [×2], C3×D12, C3×D12 [×4], C3×C3⋊D4 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, C2×S32 [×4], S3×C2×C6 [×2], D46D6, D4○D12, D125S3 [×2], D12⋊S3 [×2], S3×D12 [×2], D6⋊D6 [×2], S3×C3⋊D4 [×4], C6×D12, C3×C4○D12, C12.59D6, D1224D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D46D6, D4○D12, C22×S32, D1224D6

Smallest permutation representation of D1224D6
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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)
(1 30 5 26 9 34)(2 29 6 25 10 33)(3 28 7 36 11 32)(4 27 8 35 12 31)(13 47 17 43 21 39)(14 46 18 42 22 38)(15 45 19 41 23 37)(16 44 20 40 24 48)
(1 20)(2 13)(3 18)(4 23)(5 16)(6 21)(7 14)(8 19)(9 24)(10 17)(11 22)(12 15)(25 43)(26 48)(27 41)(28 46)(29 39)(30 44)(31 37)(32 42)(33 47)(34 40)(35 45)(36 38)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C ··· 2H 2I 2J 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I ··· 6N 12A 12B 12C ··· 12I 12J 12K 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 12 12 12 ··· 12 12 12 size 1 1 2 6 ··· 6 18 18 2 2 4 2 2 6 6 18 18 2 2 2 2 4 4 4 4 12 ··· 12 2 2 4 ··· 4 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 2+ 1+4 S32 C2×S32 C2×S32 D4⋊6D6 D4○D12 D12⋊24D6 kernel D12⋊24D6 D12⋊5S3 D12⋊S3 S3×D12 D6⋊D6 S3×C3⋊D4 C6×D12 C3×C4○D12 C12.59D6 C2×D12 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C22×S3 C32 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 2 2 2 4 1 1 1 1 1 1 2 5 2 2 2 1 1 2 1 2 2 4

Matrix representation of D1224D6 in GL4(𝔽13) generated by

 5 5 0 0 8 0 0 0 0 0 8 8 0 0 5 0
,
 0 0 8 8 0 0 0 5 5 5 0 0 0 8 0 0
,
 0 0 10 3 0 0 6 3 6 3 0 0 10 7 0 0
,
 6 3 0 0 10 7 0 0 0 0 10 3 0 0 6 3
`G:=sub<GL(4,GF(13))| [5,8,0,0,5,0,0,0,0,0,8,5,0,0,8,0],[0,0,5,0,0,0,5,8,8,0,0,0,8,5,0,0],[0,0,6,10,0,0,3,7,10,6,0,0,3,3,0,0],[6,10,0,0,3,7,0,0,0,0,10,6,0,0,3,3] >;`

D1224D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{24}D_6`
`% in TeX`

`G:=Group("D12:24D6");`
`// GroupNames label`

`G:=SmallGroup(288,955);`
`// by ID`

`G=gap.SmallGroup(288,955);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^4*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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