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G = Dic17order 68 = 22·17

Dicyclic group

Aliases: Dic17, C172C4, 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 >

Character table of Dic17

 class 1 2 4A 4B 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 i -i 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 symplectic faithful, Schur index 2 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 symplectic faithful, Schur index 2 ρ15 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 symplectic faithful, Schur index 2 ρ16 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 symplectic faithful, Schur index 2 ρ17 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 symplectic faithful, Schur index 2 ρ18 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 symplectic faithful, Schur index 2 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 symplectic faithful, Schur index 2 ρ20 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 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

 6 1 136 0
,
 9 136 82 128
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

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