Copied to
clipboard

G = D60order 120 = 23·3·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C30 — D60
C1C5C15C30D30 — D60
C15C30 — D60
C1C2C4

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 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

׿
×
𝔽