direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
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
| C17 — D34 |
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 |
►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
Subgroup lattice of D34 in TeX
Character table of D34 in TeX