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## G = D35order 70 = 2·5·7

### Dihedral group

Aliases: D35, C7⋊D5, C5⋊D7, C351C2, sometimes denoted D70 or Dih35 or Dih70, SmallGroup(70,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D35
 Chief series C1 — C7 — C35 — D35
 Lower central C35 — D35
 Upper central C1

Generators and relations for D35
G = < a,b | a35=b2=1, bab=a-1 >

Character table of D35

 class 1 2 5A 5B 7A 7B 7C 35A 35B 35C 35D 35E 35F 35G 35H 35I 35J 35K 35L size 1 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 -1-√5/2 -1+√5/2 2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ4 2 0 -1+√5/2 -1-√5/2 2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ5 2 0 2 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ6 2 0 2 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ7 2 0 2 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ8 2 0 -1-√5/2 -1+√5/2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ54ζ7+ζ5ζ76 ζ54ζ76+ζ5ζ7 ζ54ζ74+ζ5ζ73 ζ54ζ72+ζ5ζ75 ζ54ζ75+ζ5ζ72 ζ53ζ75+ζ52ζ72 ζ53ζ73+ζ52ζ74 ζ53ζ7+ζ52ζ76 ζ53ζ76+ζ52ζ7 ζ53ζ74+ζ52ζ73 ζ53ζ72+ζ52ζ75 ζ54ζ73+ζ5ζ74 orthogonal faithful ρ9 2 0 -1+√5/2 -1-√5/2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ53ζ76+ζ52ζ7 ζ53ζ7+ζ52ζ76 ζ53ζ73+ζ52ζ74 ζ53ζ75+ζ52ζ72 ζ53ζ72+ζ52ζ75 ζ54ζ75+ζ5ζ72 ζ54ζ73+ζ5ζ74 ζ54ζ7+ζ5ζ76 ζ54ζ76+ζ5ζ7 ζ54ζ74+ζ5ζ73 ζ54ζ72+ζ5ζ75 ζ53ζ74+ζ52ζ73 orthogonal faithful ρ10 2 0 -1-√5/2 -1+√5/2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ54ζ76+ζ5ζ7 ζ54ζ7+ζ5ζ76 ζ54ζ73+ζ5ζ74 ζ54ζ75+ζ5ζ72 ζ54ζ72+ζ5ζ75 ζ53ζ72+ζ52ζ75 ζ53ζ74+ζ52ζ73 ζ53ζ76+ζ52ζ7 ζ53ζ7+ζ52ζ76 ζ53ζ73+ζ52ζ74 ζ53ζ75+ζ52ζ72 ζ54ζ74+ζ5ζ73 orthogonal faithful ρ11 2 0 -1+√5/2 -1-√5/2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ53ζ74+ζ52ζ73 ζ53ζ73+ζ52ζ74 ζ53ζ72+ζ52ζ75 ζ53ζ7+ζ52ζ76 ζ53ζ76+ζ52ζ7 ζ54ζ7+ζ5ζ76 ζ54ζ72+ζ5ζ75 ζ54ζ73+ζ5ζ74 ζ54ζ74+ζ5ζ73 ζ54ζ75+ζ5ζ72 ζ54ζ76+ζ5ζ7 ζ53ζ75+ζ52ζ72 orthogonal faithful ρ12 2 0 -1-√5/2 -1+√5/2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ54ζ73+ζ5ζ74 ζ54ζ74+ζ5ζ73 ζ54ζ75+ζ5ζ72 ζ54ζ76+ζ5ζ7 ζ54ζ7+ζ5ζ76 ζ53ζ7+ζ52ζ76 ζ53ζ72+ζ52ζ75 ζ53ζ73+ζ52ζ74 ζ53ζ74+ζ52ζ73 ζ53ζ75+ζ52ζ72 ζ53ζ76+ζ52ζ7 ζ54ζ72+ζ5ζ75 orthogonal faithful ρ13 2 0 -1-√5/2 -1+√5/2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ54ζ75+ζ5ζ72 ζ54ζ72+ζ5ζ75 ζ54ζ76+ζ5ζ7 ζ54ζ73+ζ5ζ74 ζ54ζ74+ζ5ζ73 ζ53ζ74+ζ52ζ73 ζ53ζ7+ζ52ζ76 ζ53ζ75+ζ52ζ72 ζ53ζ72+ζ52ζ75 ζ53ζ76+ζ52ζ7 ζ53ζ73+ζ52ζ74 ζ54ζ7+ζ5ζ76 orthogonal faithful ρ14 2 0 -1+√5/2 -1-√5/2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ53ζ75+ζ52ζ72 ζ53ζ72+ζ52ζ75 ζ53ζ76+ζ52ζ7 ζ53ζ73+ζ52ζ74 ζ53ζ74+ζ52ζ73 ζ54ζ73+ζ5ζ74 ζ54ζ76+ζ5ζ7 ζ54ζ72+ζ5ζ75 ζ54ζ75+ζ5ζ72 ζ54ζ7+ζ5ζ76 ζ54ζ74+ζ5ζ73 ζ53ζ7+ζ52ζ76 orthogonal faithful ρ15 2 0 -1-√5/2 -1+√5/2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ54ζ74+ζ5ζ73 ζ54ζ73+ζ5ζ74 ζ54ζ72+ζ5ζ75 ζ54ζ7+ζ5ζ76 ζ54ζ76+ζ5ζ7 ζ53ζ76+ζ52ζ7 ζ53ζ75+ζ52ζ72 ζ53ζ74+ζ52ζ73 ζ53ζ73+ζ52ζ74 ζ53ζ72+ζ52ζ75 ζ53ζ7+ζ52ζ76 ζ54ζ75+ζ5ζ72 orthogonal faithful ρ16 2 0 -1+√5/2 -1-√5/2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ53ζ72+ζ52ζ75 ζ53ζ75+ζ52ζ72 ζ53ζ7+ζ52ζ76 ζ53ζ74+ζ52ζ73 ζ53ζ73+ζ52ζ74 ζ54ζ74+ζ5ζ73 ζ54ζ7+ζ5ζ76 ζ54ζ75+ζ5ζ72 ζ54ζ72+ζ5ζ75 ζ54ζ76+ζ5ζ7 ζ54ζ73+ζ5ζ74 ζ53ζ76+ζ52ζ7 orthogonal faithful ρ17 2 0 -1+√5/2 -1-√5/2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ53ζ7+ζ52ζ76 ζ53ζ76+ζ52ζ7 ζ53ζ74+ζ52ζ73 ζ53ζ72+ζ52ζ75 ζ53ζ75+ζ52ζ72 ζ54ζ72+ζ5ζ75 ζ54ζ74+ζ5ζ73 ζ54ζ76+ζ5ζ7 ζ54ζ7+ζ5ζ76 ζ54ζ73+ζ5ζ74 ζ54ζ75+ζ5ζ72 ζ53ζ73+ζ52ζ74 orthogonal faithful ρ18 2 0 -1-√5/2 -1+√5/2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ54ζ72+ζ5ζ75 ζ54ζ75+ζ5ζ72 ζ54ζ7+ζ5ζ76 ζ54ζ74+ζ5ζ73 ζ54ζ73+ζ5ζ74 ζ53ζ73+ζ52ζ74 ζ53ζ76+ζ52ζ7 ζ53ζ72+ζ52ζ75 ζ53ζ75+ζ52ζ72 ζ53ζ7+ζ52ζ76 ζ53ζ74+ζ52ζ73 ζ54ζ76+ζ5ζ7 orthogonal faithful ρ19 2 0 -1+√5/2 -1-√5/2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ53ζ73+ζ52ζ74 ζ53ζ74+ζ52ζ73 ζ53ζ75+ζ52ζ72 ζ53ζ76+ζ52ζ7 ζ53ζ7+ζ52ζ76 ζ54ζ76+ζ5ζ7 ζ54ζ75+ζ5ζ72 ζ54ζ74+ζ5ζ73 ζ54ζ73+ζ5ζ74 ζ54ζ72+ζ5ζ75 ζ54ζ7+ζ5ζ76 ζ53ζ72+ζ52ζ75 orthogonal faithful

Smallest permutation representation of D35
On 35 points
Generators in S35
```(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)```

`G:=sub<Sym(35)| (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)>;`

`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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19) );`

`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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19)]])`

D35 is a maximal subgroup of   D5×D7  C5⋊F7  D105  D175  C5⋊D35  D245  C7⋊D35
D35 is a maximal quotient of   Dic35  D105  D175  C5⋊D35  D245  C7⋊D35

Matrix representation of D35 in GL2(𝔽71) generated by

 4 41 30 6
,
 4 41 36 67
`G:=sub<GL(2,GF(71))| [4,30,41,6],[4,36,41,67] >;`

D35 in GAP, Magma, Sage, TeX

`D_{35}`
`% in TeX`

`G:=Group("D35");`
`// GroupNames label`

`G:=SmallGroup(70,3);`
`// by ID`

`G=gap.SmallGroup(70,3);`
`# by ID`

`G:=PCGroup([3,-2,-5,-7,49,542]);`
`// Polycyclic`

`G:=Group<a,b|a^35=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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