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G = D34order 68 = 22·17

Dihedral group

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

C1C17 — D34
C1C17D17 — D34
C17 — D34
C1C2

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

17C2
17C2
17C22

Character table of D34

 class 12A2B2C17A17B17C17D17E17F17G17H34A34B34C34D34E34F34G34H
 size 1117172222222222222222
ρ111111111111111111111    trivial
ρ21-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-1-11111111111111111    linear of order 2
ρ52-200ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173ζ1713174171317417111761716171791781715172171217517101771714173    orthogonal faithful
ρ62200ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174ζ1711176ζ1711176ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174    orthogonal lifted from D17
ρ72200ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173ζ1713174ζ1713174ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173    orthogonal lifted from D17
ρ82-200ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178ζ1712175171217517161717141731710177171117617151721713174179178    orthogonal faithful
ρ92-200ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175ζ171617171617171017717131741715172179178171417317111761712175    orthogonal faithful
ρ102200ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177ζ1715172ζ1715172ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177    orthogonal lifted from D17
ρ112200ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172ζ1714173ζ1714173ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172    orthogonal lifted from D17
ρ122-200ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174ζ1711176171117617917817101771712175171417317161717151721713174    orthogonal faithful
ρ132-200ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172ζ1714173171417317131741712175171117617101771791781716171715172    orthogonal faithful
ρ142200ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175ζ171617ζ171617ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175    orthogonal lifted from D17
ρ152200ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176ζ179178ζ179178ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176    orthogonal lifted from D17
ρ162-200ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176ζ179178179178171217517151721716171713174171017717141731711176    orthogonal faithful
ρ172-200ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177ζ1715172171517217141731791781713174171617171117617121751710177    orthogonal faithful
ρ182200ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617ζ1710177ζ1710177ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617    orthogonal lifted from D17
ρ192-200ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617ζ1710177171017717151721711176171417317121751713174179178171617    orthogonal faithful
ρ202200ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178ζ1712175ζ1712175ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178    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

6690
4145
,
9632
507
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

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