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## G = D30⋊S3order 360 = 23·32·5

### 3rd semidirect product of D30 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — D30⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — D30⋊S3
 Lower central C3×C15 — C3×C30 — D30⋊S3
 Upper central C1 — C2

Generators and relations for D30⋊S3
G = < a,b,c,d | a30=b2=c3=d2=1, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a25b, dcd=c-1 >

Subgroups: 444 in 74 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], C5, S3 [×5], C6 [×2], C6 [×2], D4, C32, D5, C10, C10, Dic3, C12, D6 [×4], C2×C6, C15 [×2], C15, C3×S3, C3⋊S3, C3×C6, Dic5, D10, C2×C10, D12, C3⋊D4, C5×S3 [×4], C3×D5, D15, C30 [×2], C30, C3×Dic3, S3×C6, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5, Dic15, C6×D5, S3×C10 [×3], D30, C3⋊D12, C3×D15, C5×C3⋊S3, C3×C30, C15⋊D4, C5⋊D12, C3×Dic15, C6×D15, C10×C3⋊S3, D30⋊S3
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D5, D6 [×2], D10, D12, C3⋊D4, S32, C5⋊D4, S3×D5 [×2], C3⋊D12, C15⋊D4, C5⋊D12, D15⋊S3, D30⋊S3

Smallest permutation representation of D30⋊S3
On 60 points
Generators in S60
```(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 49 50 51 52 53 54 55 56 57 58 59 60)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 36)(32 47)(33 58)(34 39)(35 50)(37 42)(38 53)(40 45)(41 56)(43 48)(44 59)(46 51)(49 54)(52 57)(55 60)```

`G:=sub<Sym(60)| (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,28)(9,19,29)(10,20,30)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,36)(32,47)(33,58)(34,39)(35,50)(37,42)(38,53)(40,45)(41,56)(43,48)(44,59)(46,51)(49,54)(52,57)(55,60)>;`

`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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,28)(9,19,29)(10,20,30)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,36)(32,47)(33,58)(34,39)(35,50)(37,42)(38,53)(40,45)(41,56)(43,48)(44,59)(46,51)(49,54)(52,57)(55,60) );`

`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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38)], [(1,11,21),(2,12,22),(3,13,23),(4,14,24),(5,15,25),(6,16,26),(7,17,27),(8,18,28),(9,19,29),(10,20,30),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(2,12),(3,23),(5,15),(6,26),(8,18),(9,29),(11,21),(14,24),(17,27),(20,30),(31,36),(32,47),(33,58),(34,39),(35,50),(37,42),(38,53),(40,45),(41,56),(43,48),(44,59),(46,51),(49,54),(52,57),(55,60)])`

39 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 5A 5B 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E 10F 12A 12B 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 3 4 5 5 6 6 6 6 6 10 10 10 10 10 10 12 12 15 ··· 15 30 ··· 30 size 1 1 18 30 2 2 4 30 2 2 2 2 4 30 30 2 2 18 18 18 18 30 30 4 ··· 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + image C1 C2 C2 C2 S3 S3 D4 D5 D6 D10 D12 C3⋊D4 C5⋊D4 S32 S3×D5 C3⋊D12 C15⋊D4 C5⋊D12 D15⋊S3 D30⋊S3 kernel D30⋊S3 C3×Dic15 C6×D15 C10×C3⋊S3 Dic15 D30 C3×C15 C2×C3⋊S3 C30 C3×C6 C15 C15 C32 C10 C6 C5 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 2 2 4 1 4 1 2 2 4 4

Matrix representation of D30⋊S3 in GL8(𝔽61)

 43 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 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 0 60 0 0 0 0 0 0 1 60
,
 56 23 0 0 0 0 0 0 52 5 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 60
,
 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 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 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 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(61))| [43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[56,52,0,0,0,0,0,0,23,5,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

D30⋊S3 in GAP, Magma, Sage, TeX

`D_{30}\rtimes S_3`
`% in TeX`

`G:=Group("D30:S3");`
`// GroupNames label`

`G:=SmallGroup(360,86);`
`// by ID`

`G=gap.SmallGroup(360,86);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,387,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^25*b,d*c*d=c^-1>;`
`// generators/relations`

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