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G = D80  order 160 = 25·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D80, C5⋊1D16, C80⋊1C2, C16⋊1D5, D40⋊1C2, C4.1D20, C2.3D40, C10.1D8, C20.24D4, C8.13D10, C40.14C22, sometimes denoted D160 or Dih80 or Dih160, SmallGroup(160,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C40 — D80
Chief series
C1 — C5 — C10 — C20 — C40 — D40 — D80
Lower central
C5 — C10 — C20 — C40 — D80
Upper central
C1 — C2 — C4 — C8 — C16

Generators and relations for D80
 G = < a,b | a80=b2=1, bab=a-1 >

C1
C2
40C2
40C2
C5
C4
20C22
20C22
C10
8D5
8D5
C8
10D4
10D4
C20
4D10
4D10
C16
5D8
5D8
C40
2D20
2D20
5D16
D40
D40
C80
D80

Smallest permutation representation of D80
►On 80 points
Generators in S80
(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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)
 
G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41)]])
 

D80 is a maximal subgroup of
 D160  C160⋊C2  C5⋊D32  C5⋊SD64  D80⋊7C2  D80⋊C2  D5×D16  C16⋊D10  D80⋊5C2  C3⋊D80  D240
D80 is a maximal quotient of
 D160  C160⋊C2  Dic80  C80⋊13C4  D40⋊7C4  C3⋊D80  D240

43 conjugacy classes

class 1 2A2B2C 4 5A5B8A8B10A10B16A16B16C16D20A20B20C20D40A···40H80A···80P
order1222455881010161616162020202040···4080···80
size1140402222222222222222···22···2

43 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2D4D5D8D10D16D20D40D80
kernelD80C80D40C20C16C10C8C5C4C2C1
# reps112122244816

Matrix representation of D80 ►in GL2(𝔽241) generated by

239173
230228
,
183102
4558
G:=sub<GL(2,GF(241))| [239,230,173,228],[183,45,102,58] >;
 

D80 in GAP, Magma, Sage, TeX 

D_{80}
 
% in TeX
 
G:=Group("D80");
 
// GroupNames label
 
G:=SmallGroup(160,6);
 
// by ID
 
G=gap.SmallGroup(160,6);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,79,218,122,579,69,4613]);
 
// Polycyclic
 
G:=Group<a,b|a^80=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D80 in TeX

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