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G = D90  order 180 = 22·32·5

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D90, C2×D45, C10⋊D9, C18⋊D5, C5⋊2D18, C9⋊2D10, C3.D30, C90⋊1C2, C30.2S3, C45⋊2C22, C6.2D15, C15.2D6, sometimes denoted D180 or Dih90 or Dih180, SmallGroup(180,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C45 — D90
Chief series
C1 — C3 — C15 — C45 — D45 — D90
Lower central
C45 — D90
Upper central
C1 — C2

Generators and relations for D90
 G = < a,b | a90=b2=1, bab=a-1 >

C1
C2
45C2
45C2
C3
C5
45C22
C6
15S3
15S3
C9
C10
9D5
9D5
C15
15D6
C18
5D9
5D9
9D10
C30
3D15
3D15
C45
5D18
3D30
D45
C90
D45
D90

Smallest permutation representation of D90
►On 90 points
Generators in S90
(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 81 82 83 84 85 86 87 88 89 90)

(1 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)
 
G:=sub<Sym(90)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46)]])
 

D90 is a maximal subgroup of   D90.C2  C5⋊D36  C9⋊D20  D180  C45⋊7D4  C2×D5×D9
D90 is a maximal quotient of   Dic90  D180  C45⋊7D4

48 conjugacy classes

class 1 2A2B2C 3 5A5B 6 9A9B9C10A10B15A15B15C15D18A18B18C30A30B30C30D45A···45L90A···90L
order122235569991010151515151818183030303045···4590···90
size114545222222222222222222222···22···2

48 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D5D6D9D10D15D18D30D45D90
kernelD90D45C90C30C18C15C10C9C6C5C3C2C1
# reps121121324341212

Matrix representation of D90 ►in GL4(𝔽181) generated by

11400
16716700
0088129
0052140
,
11400
018000
0088129
004193
G:=sub<GL(4,GF(181))| [1,167,0,0,14,167,0,0,0,0,88,52,0,0,129,140],[1,0,0,0,14,180,0,0,0,0,88,41,0,0,129,93] >;
 

D90 in GAP, Magma, Sage, TeX 

D_{90}
 
% in TeX
 
G:=Group("D90");
 
// GroupNames label
 
G:=SmallGroup(180,11);
 
// by ID
 
G=gap.SmallGroup(180,11);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-5,-3,1022,462,963,3004]);
 
// Polycyclic
 
G:=Group<a,b|a^90=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D90 in TeX

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