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## G = C3⋊D20order 120 = 23·3·5

### The semidirect product of C3 and D20 acting via D20/D10=C2

Aliases: C152D4, C32D20, Dic3⋊D5, D102S3, D303C2, C6.5D10, C10.5D6, C30.5C22, (C6×D5)⋊2C2, C51(C3⋊D4), C2.5(S3×D5), (C5×Dic3)⋊3C2, SmallGroup(120,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3⋊D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C3⋊D20
 Lower central C15 — C30 — C3⋊D20
 Upper central C1 — C2

Generators and relations for C3⋊D20
G = < a,b,c | a3=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D20

 class 1 2A 2B 2C 3 4 5A 5B 6A 6B 6C 10A 10B 15A 15B 20A 20B 20C 20D 30A 30B size 1 1 10 30 2 6 2 2 2 10 10 2 2 4 4 6 6 6 6 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 -2 0 0 2 0 2 2 -2 0 0 -2 -2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 0 -1 0 2 2 -1 -1 -1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 0 -1 0 2 2 -1 1 1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ8 2 2 0 0 2 2 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 0 0 2 2 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 0 0 2 -2 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ11 2 2 0 0 2 -2 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ12 2 -2 0 0 2 0 -1-√5/2 -1+√5/2 -2 0 0 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 1+√5/2 1-√5/2 orthogonal lifted from D20 ρ13 2 -2 0 0 2 0 -1+√5/2 -1-√5/2 -2 0 0 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 1-√5/2 1+√5/2 orthogonal lifted from D20 ρ14 2 -2 0 0 2 0 -1-√5/2 -1+√5/2 -2 0 0 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 1+√5/2 1-√5/2 orthogonal lifted from D20 ρ15 2 -2 0 0 2 0 -1+√5/2 -1-√5/2 -2 0 0 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 1-√5/2 1+√5/2 orthogonal lifted from D20 ρ16 2 -2 0 0 -1 0 2 2 1 -√-3 √-3 -2 -2 -1 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ17 2 -2 0 0 -1 0 2 2 1 √-3 -√-3 -2 -2 -1 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ18 4 -4 0 0 -2 0 -1-√5 -1+√5 2 0 0 1-√5 1+√5 1+√5/2 1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 orthogonal faithful, Schur index 2 ρ19 4 4 0 0 -2 0 -1-√5 -1+√5 -2 0 0 -1+√5 -1-√5 1+√5/2 1-√5/2 0 0 0 0 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ20 4 4 0 0 -2 0 -1+√5 -1-√5 -2 0 0 -1-√5 -1+√5 1-√5/2 1+√5/2 0 0 0 0 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ21 4 -4 0 0 -2 0 -1+√5 -1-√5 2 0 0 1+√5 1-√5 1-√5/2 1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 orthogonal faithful, Schur index 2

Smallest permutation representation of C3⋊D20
On 60 points
Generators in S60
```(1 24 57)(2 58 25)(3 26 59)(4 60 27)(5 28 41)(6 42 29)(7 30 43)(8 44 31)(9 32 45)(10 46 33)(11 34 47)(12 48 35)(13 36 49)(14 50 37)(15 38 51)(16 52 39)(17 40 53)(18 54 21)(19 22 55)(20 56 23)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)```

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

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

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

C3⋊D20 is a maximal subgroup of
D20⋊S3  D6.D10  C12.28D10  S3×D20  Dic5.D6  D5×C3⋊D4  D10⋊D6  C9⋊D20  C327D20  C3⋊D60  C323D20  D10.2S4  A4⋊D20
C3⋊D20 is a maximal quotient of
C3⋊D40  C6.D20  C15⋊SD16  C3⋊Dic20  D10⋊Dic3  D304C4  C6.Dic10  C9⋊D20  C327D20  C3⋊D60  C323D20  A4⋊D20

Matrix representation of C3⋊D20 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 1 46 0 0 49 59
,
 7 32 0 0 29 2 0 0 0 0 27 35 0 0 14 34
,
 0 60 0 0 60 0 0 0 0 0 60 0 0 0 12 1
`G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,49,0,0,46,59],[7,29,0,0,32,2,0,0,0,0,27,14,0,0,35,34],[0,60,0,0,60,0,0,0,0,0,60,12,0,0,0,1] >;`

C3⋊D20 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{20}`
`% in TeX`

`G:=Group("C3:D20");`
`// GroupNames label`

`G:=SmallGroup(120,12);`
`// by ID`

`G=gap.SmallGroup(120,12);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-5,61,26,168,2404]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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