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## G = D90order 180 = 22·32·5

### Dihedral group

Aliases: D90, C2×D45, C10⋊D9, C18⋊D5, C52D18, C92D10, C3.D30, C901C2, C30.2S3, C452C22, 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 >

45C2
45C2
45C22
15S3
15S3
9D5
9D5
15D6
5D9
5D9
9D10
3D15
3D15
5D18
3D30

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  C457D4  C2×D5×D9
D90 is a maximal quotient of   Dic90  D180  C457D4

48 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 6 9A 9B 9C 10A 10B 15A 15B 15C 15D 18A 18B 18C 30A 30B 30C 30D 45A ··· 45L 90A ··· 90L order 1 2 2 2 3 5 5 6 9 9 9 10 10 15 15 15 15 18 18 18 30 30 30 30 45 ··· 45 90 ··· 90 size 1 1 45 45 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 D5 D6 D9 D10 D15 D18 D30 D45 D90 kernel D90 D45 C90 C30 C18 C15 C10 C9 C6 C5 C3 C2 C1 # reps 1 2 1 1 2 1 3 2 4 3 4 12 12

Matrix representation of D90 in GL4(𝔽181) generated by

 1 14 0 0 167 167 0 0 0 0 88 129 0 0 52 140
,
 1 14 0 0 0 180 0 0 0 0 88 129 0 0 41 93
`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`

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