Copied to
clipboard

## G = D34order 68 = 22·17

### Dihedral group

Aliases: D34, C2×D17, C34⋊C2, C17⋊C22, sometimes denoted D68 or Dih34 or Dih68, SmallGroup(68,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — D34
 Chief series C1 — C17 — D17 — D34
 Lower central C17 — D34
 Upper central C1 — C2

Generators and relations for D34
G = < a,b | a34=b2=1, bab=a-1 >

Character table of D34

 class 1 2A 2B 2C 17A 17B 17C 17D 17E 17F 17G 17H 34A 34B 34C 34D 34E 34F 34G 34H size 1 1 17 17 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 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 -2 0 0 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 ζ1713+ζ174 -ζ1713-ζ174 -ζ1711-ζ176 -ζ1716-ζ17 -ζ179-ζ178 -ζ1715-ζ172 -ζ1712-ζ175 -ζ1710-ζ177 -ζ1714-ζ173 orthogonal faithful ρ6 2 2 0 0 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 ζ1711+ζ176 ζ1711+ζ176 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 orthogonal lifted from D17 ρ7 2 2 0 0 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 ζ1713+ζ174 ζ1713+ζ174 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 orthogonal lifted from D17 ρ8 2 -2 0 0 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 ζ1712+ζ175 -ζ1712-ζ175 -ζ1716-ζ17 -ζ1714-ζ173 -ζ1710-ζ177 -ζ1711-ζ176 -ζ1715-ζ172 -ζ1713-ζ174 -ζ179-ζ178 orthogonal faithful ρ9 2 -2 0 0 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 ζ1716+ζ17 -ζ1716-ζ17 -ζ1710-ζ177 -ζ1713-ζ174 -ζ1715-ζ172 -ζ179-ζ178 -ζ1714-ζ173 -ζ1711-ζ176 -ζ1712-ζ175 orthogonal faithful ρ10 2 2 0 0 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 ζ1715+ζ172 ζ1715+ζ172 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 orthogonal lifted from D17 ρ11 2 2 0 0 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 ζ1714+ζ173 ζ1714+ζ173 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 orthogonal lifted from D17 ρ12 2 -2 0 0 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 ζ1711+ζ176 -ζ1711-ζ176 -ζ179-ζ178 -ζ1710-ζ177 -ζ1712-ζ175 -ζ1714-ζ173 -ζ1716-ζ17 -ζ1715-ζ172 -ζ1713-ζ174 orthogonal faithful ρ13 2 -2 0 0 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 ζ1714+ζ173 -ζ1714-ζ173 -ζ1713-ζ174 -ζ1712-ζ175 -ζ1711-ζ176 -ζ1710-ζ177 -ζ179-ζ178 -ζ1716-ζ17 -ζ1715-ζ172 orthogonal faithful ρ14 2 2 0 0 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 ζ1716+ζ17 ζ1716+ζ17 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 orthogonal lifted from D17 ρ15 2 2 0 0 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 ζ179+ζ178 ζ179+ζ178 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 orthogonal lifted from D17 ρ16 2 -2 0 0 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 ζ179+ζ178 -ζ179-ζ178 -ζ1712-ζ175 -ζ1715-ζ172 -ζ1716-ζ17 -ζ1713-ζ174 -ζ1710-ζ177 -ζ1714-ζ173 -ζ1711-ζ176 orthogonal faithful ρ17 2 -2 0 0 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 ζ1715+ζ172 -ζ1715-ζ172 -ζ1714-ζ173 -ζ179-ζ178 -ζ1713-ζ174 -ζ1716-ζ17 -ζ1711-ζ176 -ζ1712-ζ175 -ζ1710-ζ177 orthogonal faithful ρ18 2 2 0 0 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 ζ1710+ζ177 ζ1710+ζ177 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 orthogonal lifted from D17 ρ19 2 -2 0 0 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 ζ1710+ζ177 -ζ1710-ζ177 -ζ1715-ζ172 -ζ1711-ζ176 -ζ1714-ζ173 -ζ1712-ζ175 -ζ1713-ζ174 -ζ179-ζ178 -ζ1716-ζ17 orthogonal faithful ρ20 2 2 0 0 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 ζ1712+ζ175 ζ1712+ζ175 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 orthogonal lifted from D17

Smallest permutation representation of D34
On 34 points
Generators in S34
```(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)
(1 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 18)```

`G:=sub<Sym(34)| (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)>;`

`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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18) );`

`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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,18)])`

D34 is a maximal subgroup of   D68  C17⋊D4
D34 is a maximal quotient of   Dic34  D68  C17⋊D4

Matrix representation of D34 in GL2(𝔽103) generated by

 66 90 41 45
,
 96 32 50 7
`G:=sub<GL(2,GF(103))| [66,41,90,45],[96,50,32,7] >;`

D34 in GAP, Magma, Sage, TeX

`D_{34}`
`% in TeX`

`G:=Group("D34");`
`// GroupNames label`

`G:=SmallGroup(68,4);`
`// by ID`

`G=gap.SmallGroup(68,4);`
`# by ID`

`G:=PCGroup([3,-2,-2,-17,578]);`
`// Polycyclic`

`G:=Group<a,b|a^34=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽