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G = Dic17  order 68 = 22·17

Dicyclic group

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

Aliases: Dic17, C17⋊2C4, C34.C2, C2.D17, SmallGroup(68,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C17 — Dic17
Chief series
C1 — C17 — C34 — Dic17
Lower central
C17 — Dic17
Upper central
C1 — C2

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

C1
C2
C17
17C4
C34
Dic17

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
ρ52200ζ179+ζ178ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174ζ1711+ζ176ζ1711+ζ176ζ179+ζ178ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174    orthogonal lifted from D17
ρ62200ζ1711+ζ176ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173ζ1713+ζ174ζ1713+ζ174ζ1711+ζ176ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173    orthogonal lifted from D17
ρ72200ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1715+ζ172ζ1713+ζ174ζ179+ζ178ζ1712+ζ175ζ1712+ζ175ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1715+ζ172ζ1713+ζ174ζ179+ζ178    orthogonal lifted from D17
ρ82200ζ1714+ζ173ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177ζ1715+ζ172ζ1715+ζ172ζ1714+ζ173ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177    orthogonal lifted from D17
ρ92200ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ179+ζ178ζ1716+ζ17ζ1715+ζ172ζ1714+ζ173ζ1714+ζ173ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ179+ζ178ζ1716+ζ17ζ1715+ζ172    orthogonal lifted from D17
ρ102200ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175ζ1716+ζ17ζ1716+ζ17ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175    orthogonal lifted from D17
ρ112200ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176ζ179+ζ178ζ179+ζ178ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176    orthogonal lifted from D17
ρ122200ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1713+ζ174ζ179+ζ178ζ1716+ζ17ζ1710+ζ177ζ1710+ζ177ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1713+ζ174ζ179+ζ178ζ1716+ζ17    orthogonal lifted from D17
ρ132-200ζ1711+ζ176ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173ζ1713+ζ174-ζ1713-ζ174-ζ1711-ζ176-ζ1716-ζ17-ζ179-ζ178-ζ1715-ζ172-ζ1712-ζ175-ζ1710-ζ177-ζ1714-ζ173    symplectic faithful, Schur index 2
ρ142-200ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175ζ1716+ζ17-ζ1716-ζ17-ζ1710-ζ177-ζ1713-ζ174-ζ1715-ζ172-ζ179-ζ178-ζ1714-ζ173-ζ1711-ζ176-ζ1712-ζ175    symplectic faithful, Schur index 2
ρ152-200ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1715+ζ172ζ1713+ζ174ζ179+ζ178ζ1712+ζ175-ζ1712-ζ175-ζ1716-ζ17-ζ1714-ζ173-ζ1710-ζ177-ζ1711-ζ176-ζ1715-ζ172-ζ1713-ζ174-ζ179-ζ178    symplectic faithful, Schur index 2
ρ162-200ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ179+ζ178ζ1716+ζ17ζ1715+ζ172ζ1714+ζ173-ζ1714-ζ173-ζ1713-ζ174-ζ1712-ζ175-ζ1711-ζ176-ζ1710-ζ177-ζ179-ζ178-ζ1716-ζ17-ζ1715-ζ172    symplectic faithful, Schur index 2
ρ172-200ζ179+ζ178ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174ζ1711+ζ176-ζ1711-ζ176-ζ179-ζ178-ζ1710-ζ177-ζ1712-ζ175-ζ1714-ζ173-ζ1716-ζ17-ζ1715-ζ172-ζ1713-ζ174    symplectic faithful, Schur index 2
ρ182-200ζ1714+ζ173ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177ζ1715+ζ172-ζ1715-ζ172-ζ1714-ζ173-ζ179-ζ178-ζ1713-ζ174-ζ1716-ζ17-ζ1711-ζ176-ζ1712-ζ175-ζ1710-ζ177    symplectic faithful, Schur index 2
ρ192-200ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1713+ζ174ζ179+ζ178ζ1716+ζ17ζ1710+ζ177-ζ1710-ζ177-ζ1715-ζ172-ζ1711-ζ176-ζ1714-ζ173-ζ1712-ζ175-ζ1713-ζ174-ζ179-ζ178-ζ1716-ζ17    symplectic faithful, Schur index 2
ρ202-200ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176ζ179+ζ178-ζ179-ζ178-ζ1712-ζ175-ζ1715-ζ172-ζ1716-ζ17-ζ1713-ζ174-ζ1710-ζ177-ζ1714-ζ173-ζ1711-ζ176    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 65 18 48)(2 64 19 47)(3 63 20 46)(4 62 21 45)(5 61 22 44)(6 60 23 43)(7 59 24 42)(8 58 25 41)(9 57 26 40)(10 56 27 39)(11 55 28 38)(12 54 29 37)(13 53 30 36)(14 52 31 35)(15 51 32 68)(16 50 33 67)(17 49 34 66)
 
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), (1,65,18,48)(2,64,19,47)(3,63,20,46)(4,62,21,45)(5,61,22,44)(6,60,23,43)(7,59,24,42)(8,58,25,41)(9,57,26,40)(10,56,27,39)(11,55,28,38)(12,54,29,37)(13,53,30,36)(14,52,31,35)(15,51,32,68)(16,50,33,67)(17,49,34,66)>;
 
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), (1,65,18,48)(2,64,19,47)(3,63,20,46)(4,62,21,45)(5,61,22,44)(6,60,23,43)(7,59,24,42)(8,58,25,41)(9,57,26,40)(10,56,27,39)(11,55,28,38)(12,54,29,37)(13,53,30,36)(14,52,31,35)(15,51,32,68)(16,50,33,67)(17,49,34,66) );
 
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)], [(1,65,18,48),(2,64,19,47),(3,63,20,46),(4,62,21,45),(5,61,22,44),(6,60,23,43),(7,59,24,42),(8,58,25,41),(9,57,26,40),(10,56,27,39),(11,55,28,38),(12,54,29,37),(13,53,30,36),(14,52,31,35),(15,51,32,68),(16,50,33,67),(17,49,34,66)]])
 

Dic17 is a maximal subgroup of
 C17⋊2C8  C4×D17  C17⋊D4  C17⋊3F5
 Dic17p: Dic34  Dic51  Dic85  Dic119 ...
Dic17 is a maximal quotient of
 C17⋊3F5
 C2p.D17: C17⋊3C8  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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