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G = D60  order 120 = 23·3·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D60, C4⋊D15, C15⋊4D4, C5⋊1D12, C3⋊1D20, C60⋊1C2, C20⋊1S3, C12⋊1D5, D30⋊1C2, 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 >

C1
C2
30C2
30C2
C3
C5
C4
15C22
15C22
C6
10S3
10S3
C10
6D5
6D5
C15
15D4
C12
5D6
5D6
C20
3D10
3D10
C30
2D15
2D15
5D12
3D20
D30
D30
C60
D60

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  Q8⋊2D15  D60⋊C2  C12.28D10  D5×D12  S3×D20  D60⋊11C2  D4×D15  Q8⋊3D15  D180  C3⋊D60  C60⋊S3  C20⋊S4  C20.3S4
D60 is a maximal quotient of
 C24⋊D5  D120  Dic60  C60⋊5C4  D30⋊3C4  D180  C3⋊D60  C60⋊S3  C20⋊S4

33 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 10A10B12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1222345561010121215151515202020203030303060···60
size1130302222222222222222222222···2

33 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D5D6D10D12D15D20D30D60
kernelD60C60D30C20C15C12C10C6C5C4C3C2C1
# reps1121121224448

Matrix representation of D60 ►in GL2(𝔽61) generated by

2855
614
,
2855
3933
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
 

Export

Subgroup lattice of D60 in TeX

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