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## G = D60order 120 = 23·3·5

### Dihedral group

Aliases: D60, C4⋊D15, C154D4, C51D12, C31D20, C601C2, C201S3, C121D5, D301C2, C2.4D30, C6.10D10, C10.10D6, C30.10C22, sometimes denoted D120 or Dih60 or Dih120, SmallGroup(120,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60
 Chief series C1 — C5 — C15 — C30 — D30 — D60
 Lower central C15 — C30 — D60
 Upper central C1 — C2 — C4

Generators and relations for D60
G = < a,b | a60=b2=1, bab=a-1 >

30C2
30C2
15C22
15C22
10S3
10S3
6D5
6D5
15D4
5D6
5D6
3D10
3D10
2D15
2D15
5D12
3D20

Smallest permutation representation of D60
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)```

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

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

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

D60 is a maximal subgroup of
C3⋊D40  C5⋊D24  C15⋊SD16  Dic6⋊D5  C24⋊D5  D120  D4⋊D15  Q82D15  D60⋊C2  C12.28D10  D5×D12  S3×D20  D6011C2  D4×D15  Q83D15  D180  C3⋊D60  C60⋊S3  C20⋊S4  C20.3S4
D60 is a maximal quotient of
C24⋊D5  D120  Dic60  C605C4  D303C4  D180  C3⋊D60  C60⋊S3  C20⋊S4

33 conjugacy classes

 class 1 2A 2B 2C 3 4 5A 5B 6 10A 10B 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 3 4 5 5 6 10 10 12 12 15 15 15 15 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 30 30 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2

33 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 D4 D5 D6 D10 D12 D15 D20 D30 D60 kernel D60 C60 D30 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 2 1 1 2 1 2 2 4 4 4 8

Matrix representation of D60 in GL2(𝔽61) generated by

 28 55 6 14
,
 28 55 39 33
`G:=sub<GL(2,GF(61))| [28,6,55,14],[28,39,55,33] >;`

D60 in GAP, Magma, Sage, TeX

`D_{60}`
`% in TeX`

`G:=Group("D60");`
`// GroupNames label`

`G:=SmallGroup(120,28);`
`// by ID`

`G=gap.SmallGroup(120,28);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-5,61,26,323,2404]);`
`// Polycyclic`

`G:=Group<a,b|a^60=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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