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## G = D85order 170 = 2·5·17

### Dihedral group

Aliases: D85, C17⋊D5, C5⋊D17, C851C2, sometimes denoted D170 or Dih85 or Dih170, SmallGroup(170,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C85 — D85
 Chief series C1 — C17 — C85 — D85
 Lower central C85 — D85
 Upper central C1

Generators and relations for D85
G = < a,b | a85=b2=1, bab=a-1 >

85C2
17D5
5D17

Smallest permutation representation of D85
On 85 points
Generators in S85
```(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)
(1 85)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)```

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

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

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

D85 is a maximal subgroup of   C17⋊F5  C852C4  D5×D17
D85 is a maximal quotient of   Dic85

44 conjugacy classes

 class 1 2 5A 5B 17A ··· 17H 85A ··· 85AF order 1 2 5 5 17 ··· 17 85 ··· 85 size 1 85 2 2 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D5 D17 D85 kernel D85 C85 C17 C5 C1 # reps 1 1 2 8 32

Matrix representation of D85 in GL2(𝔽1021) generated by

 134 383 644 233
,
 427 632 138 594
`G:=sub<GL(2,GF(1021))| [134,644,383,233],[427,138,632,594] >;`

D85 in GAP, Magma, Sage, TeX

`D_{85}`
`% in TeX`

`G:=Group("D85");`
`// GroupNames label`

`G:=SmallGroup(170,3);`
`// by ID`

`G=gap.SmallGroup(170,3);`
`# by ID`

`G:=PCGroup([3,-2,-5,-17,49,1442]);`
`// Polycyclic`

`G:=Group<a,b|a^85=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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