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## G = D24⋊5S3order 288 = 25·32

### 5th semidirect product of D24 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D24⋊5S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D24⋊5S3
 Lower central C32 — C3×C6 — C3×C12 — D24⋊5S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D245S3
G = < a,b,c,d | a24=b2=c3=d2=1, bab=a-1, ac=ca, dad=a17, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 562 in 135 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×5], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×2], C4×S3 [×5], D12 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×3], D24, Dic12, D4.S3 [×2], Q82S3 [×2], C3×D8, C3×Q16, D42S3 [×2], Q83S3 [×2], C324C8, C3×C24, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6 [×2], C3×D12 [×2], C4×C3⋊S3, D83S3, D24⋊C2, Dic6⋊S3 [×2], C3×D24, C3×Dic12, C8×C3⋊S3, D12⋊S3 [×2], D245S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, D83S3, D24⋊C2, D6⋊D6, D245S3

Smallest permutation representation of D245S3
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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 29)(26 46)(27 39)(28 32)(30 42)(31 35)(33 45)(34 38)(36 48)(37 41)(40 44)(43 47)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 12 12 18 2 2 4 2 9 9 12 12 2 2 4 24 24 2 2 18 18 4 4 4 4 24 24 4 ··· 4

36 irreducible representations

 dim 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 S3 S3 D4 D4 D6 D6 D6 C4○D8 S32 S3×D4 C2×S32 D8⋊3S3 D24⋊C2 D6⋊D6 D24⋊5S3 kernel D24⋊5S3 Dic6⋊S3 C3×D24 C3×Dic12 C8×C3⋊S3 D12⋊S3 D24 Dic12 C3⋊Dic3 C2×C3⋊S3 C24 Dic6 D12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 2 1 1 1 2 1 1 1 1 2 2 2 4 1 2 1 2 2 2 4

Matrix representation of D245S3 in GL6(𝔽73)

 22 0 0 0 0 0 18 10 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 51 39 0 0 0 0 55 22 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 3 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [22,18,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,55,0,0,0,0,39,22,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D245S3 in GAP, Magma, Sage, TeX

`D_{24}\rtimes_5S_3`
`% in TeX`

`G:=Group("D24:5S3");`
`// GroupNames label`

`G:=SmallGroup(288,458);`
`// by ID`

`G=gap.SmallGroup(288,458);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,142,675,346,80,1356,9414]);`
`// 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^17,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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