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## G = D24⋊S3order 288 = 25·32

### 2nd semidirect product of D24 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D24⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — D24⋊S3
 Lower central C32 — C3×C6 — C3×C12 — D24⋊S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D24⋊S3
G = < a,b,c,d | a24=b2=c3=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a12b, dcd=c-1 >

Subgroups: 778 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, C8, C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3, Dic3 [×3], C12 [×2], C12 [×2], 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, C24 [×2], C24 [×2], Dic6 [×3], C4×S3, C4×S3, D12 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×3], C2×C12, C3×D4 [×2], C22×S3 [×2], C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6, S3×C6 [×2], C2×C3⋊S3, C8⋊S3, C24⋊C2 [×4], D24, D24, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C2×D12, C4○D12, S3×D4, D42S3, C3×C3⋊C8, C3×C24, S3×Dic3, D6⋊S3, C3⋊D12, S3×C12, C3×D12 [×2], C324Q8, C12⋊S3, C2×S32, C8⋊D6, D8⋊S3, C3⋊D24, D12.S3, C3×C8⋊S3, C3×D24, C242S3, D125S3, S3×D12, D24⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, S3×D4, C2×S32, C8⋊D6, D8⋊S3, S3×D12, D24⋊S3

Smallest permutation representation of D24⋊S3
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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 33)(2 46)(3 35)(4 48)(5 37)(6 26)(7 39)(8 28)(9 41)(10 30)(11 43)(12 32)(13 45)(14 34)(15 47)(16 36)(17 25)(18 38)(19 27)(20 40)(21 29)(22 42)(23 31)(24 44)```

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

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

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

36 conjugacy classes

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

36 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 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D12 D12 C8⋊C22 S32 S3×D4 C2×S32 C8⋊D6 D8⋊S3 S3×D12 D24⋊S3 kernel D24⋊S3 C3⋊D24 D12.S3 C3×C8⋊S3 C3×D24 C24⋊2S3 D12⋊5S3 S3×D12 C8⋊S3 D24 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 D12 Dic3 D6 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 4

Matrix representation of D24⋊S3 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 59 7 0 0 0 0 0 0 66 66 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 66 66 0 0 0 0 0 0 59 7 0 0 0 0 0 0 0 0 6 6 32 0 0 0 0 0 6 67 0 41 0 0 0 0 32 0 67 6 0 0 0 0 0 41 6 6
,
 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 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 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 32 0 67 6 0 0 0 0 0 32 67 67 0 0 0 0 67 67 41 0 0 0 0 0 6 67 0 41

`G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,59,66,0,0,0,0,0,0,7,66,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,0,0,66,7,0,0,0,0,0,0,0,0,6,6,32,0,0,0,0,0,6,67,0,41,0,0,0,0,32,0,67,6,0,0,0,0,0,41,6,6],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,67,6,0,0,0,0,0,32,67,67,0,0,0,0,67,67,41,0,0,0,0,0,6,67,0,41] >;`

D24⋊S3 in GAP, Magma, Sage, TeX

`D_{24}\rtimes S_3`
`% in TeX`

`G:=Group("D24:S3");`
`// GroupNames label`

`G:=SmallGroup(288,443);`
`// by ID`

`G=gap.SmallGroup(288,443);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,142,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;`
`// generators/relations`

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