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G = Dic17order 68 = 22·17

Dicyclic group

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

Aliases: Dic17, C172C4, C34.C2, C2.D17, SmallGroup(68,1)

Series: Derived Chief Lower central Upper central

C1C17 — Dic17
C1C17C34 — Dic17
C17 — Dic17
C1C2

Generators and relations for Dic17
 G = < a,b | a34=1, b2=a17, bab-1=a-1 >

17C4

Character table of Dic17

 class 124A4B17A17B17C17D17E17F17G17H34A34B34C34D34E34F34G34H
 size 1117172222222222222222
ρ111111111111111111111    trivial
ρ211-1-11111111111111111    linear of order 2
ρ31-1i-i11111111-1-1-1-1-1-1-1-1    linear of order 4
ρ41-1-ii11111111-1-1-1-1-1-1-1-1    linear of order 4
ρ52200ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174ζ1711176ζ1711176ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174    orthogonal lifted from D17
ρ62200ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173ζ1713174ζ1713174ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173    orthogonal lifted from D17
ρ72200ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178ζ1712175ζ1712175ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178    orthogonal lifted from D17
ρ82200ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177ζ1715172ζ1715172ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177    orthogonal lifted from D17
ρ92200ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172ζ1714173ζ1714173ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172    orthogonal lifted from D17
ρ102200ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175ζ171617ζ171617ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175    orthogonal lifted from D17
ρ112200ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176ζ179178ζ179178ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176    orthogonal lifted from D17
ρ122200ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617ζ1710177ζ1710177ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617    orthogonal lifted from D17
ρ132-200ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177ζ1714173ζ1713174171317417111761716171791781715172171217517101771714173    symplectic faithful, Schur index 2
ρ142-200ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176ζ1712175ζ171617171617171017717131741715172179178171417317111761712175    symplectic faithful, Schur index 2
ρ152-200ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174ζ179178ζ1712175171217517161717141731710177171117617151721713174179178    symplectic faithful, Schur index 2
ρ162-200ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617ζ1715172ζ1714173171417317131741712175171117617101771791781716171715172    symplectic faithful, Schur index 2
ρ172-200ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172ζ1713174ζ1711176171117617917817101771712175171417317161717151721713174    symplectic faithful, Schur index 2
ρ182-200ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175ζ1710177ζ1715172171517217141731791781713174171617171117617121751710177    symplectic faithful, Schur index 2
ρ192-200ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178ζ171617ζ1710177171017717151721711176171417317121751713174179178171617    symplectic faithful, Schur index 2
ρ202-200ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173ζ1711176ζ179178179178171217517151721716171713174171017717141731711176    symplectic faithful, Schur index 2

Smallest permutation representation of Dic17
Regular action on 68 points
Generators in S68
(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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)
(1 49 18 66)(2 48 19 65)(3 47 20 64)(4 46 21 63)(5 45 22 62)(6 44 23 61)(7 43 24 60)(8 42 25 59)(9 41 26 58)(10 40 27 57)(11 39 28 56)(12 38 29 55)(13 37 30 54)(14 36 31 53)(15 35 32 52)(16 68 33 51)(17 67 34 50)

G:=sub<Sym(68)| (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,49,18,66)(2,48,19,65)(3,47,20,64)(4,46,21,63)(5,45,22,62)(6,44,23,61)(7,43,24,60)(8,42,25,59)(9,41,26,58)(10,40,27,57)(11,39,28,56)(12,38,29,55)(13,37,30,54)(14,36,31,53)(15,35,32,52)(16,68,33,51)(17,67,34,50)>;

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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,49,18,66)(2,48,19,65)(3,47,20,64)(4,46,21,63)(5,45,22,62)(6,44,23,61)(7,43,24,60)(8,42,25,59)(9,41,26,58)(10,40,27,57)(11,39,28,56)(12,38,29,55)(13,37,30,54)(14,36,31,53)(15,35,32,52)(16,68,33,51)(17,67,34,50) );

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),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)], [(1,49,18,66),(2,48,19,65),(3,47,20,64),(4,46,21,63),(5,45,22,62),(6,44,23,61),(7,43,24,60),(8,42,25,59),(9,41,26,58),(10,40,27,57),(11,39,28,56),(12,38,29,55),(13,37,30,54),(14,36,31,53),(15,35,32,52),(16,68,33,51),(17,67,34,50)])

Dic17 is a maximal subgroup of
C172C8  C4×D17  C17⋊D4  C173F5
 Dic17p: Dic34  Dic51  Dic85  Dic119 ...
Dic17 is a maximal quotient of
C173F5
 C2p.D17: C173C8  Dic51  Dic85  Dic119 ...

Matrix representation of Dic17 in GL2(𝔽137) generated by

61
1360
,
9136
82128
G:=sub<GL(2,GF(137))| [6,136,1,0],[9,82,136,128] >;

Dic17 in GAP, Magma, Sage, TeX

{\rm Dic}_{17}
% in TeX

G:=Group("Dic17");
// GroupNames label

G:=SmallGroup(68,1);
// by ID

G=gap.SmallGroup(68,1);
# by ID

G:=PCGroup([3,-2,-2,-17,6,578]);
// Polycyclic

G:=Group<a,b|a^34=1,b^2=a^17,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic17 in TeX
Character table of Dic17 in TeX

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