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G = D17order 34 = 2·17

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D17, C17⋊C2, sometimes denoted D34 or Dih17 or Dih34, SmallGroup(34,1)

Series: Derived Chief Lower central Upper central

C1C17 — D17
C1C17 — D17
C17 — D17
C1

Generators and relations for D17
 G = < a,b | a17=b2=1, bab=a-1 >

17C2

Character table of D17

 class 1217A17B17C17D17E17F17G17H
 size 11722222222
ρ11111111111    trivial
ρ21-111111111    linear of order 2
ρ320ζ1715172ζ1714173ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617    orthogonal faithful
ρ420ζ1710177ζ1715172ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175    orthogonal faithful
ρ520ζ179178ζ1712175ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174    orthogonal faithful
ρ620ζ1714173ζ1713174ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177    orthogonal faithful
ρ720ζ1711176ζ179178ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173    orthogonal faithful
ρ820ζ1712175ζ171617ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176    orthogonal faithful
ρ920ζ1713174ζ1711176ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172    orthogonal faithful
ρ1020ζ171617ζ1710177ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178    orthogonal faithful

Permutation representations of D17
On 17 points: primitive - transitive group 17T2
Generators in S17
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)

G:=sub<Sym(17)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)])

G:=TransitiveGroup(17,2);

Matrix representation of D17 in GL2(𝔽103) generated by

27102
10
,
27102
776
G:=sub<GL(2,GF(103))| [27,1,102,0],[27,7,102,76] >;

D17 in GAP, Magma, Sage, TeX

D_{17}
% in TeX

G:=Group("D17");
// GroupNames label

G:=SmallGroup(34,1);
// by ID

G=gap.SmallGroup(34,1);
# by ID

G:=PCGroup([2,-2,-17,129]);
// Polycyclic

G:=Group<a,b|a^17=b^2=1,b*a*b=a^-1>;
// generators/relations

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