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G = D43  order 86 = 2·43

Dihedral group

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

Aliases: D43, C43⋊C2, sometimes denoted D86 or Dih43 or Dih86, SmallGroup(86,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C43 — D43
Chief series
C1 — C43 — D43
Lower central
C43 — D43
Upper central
C1

Generators and relations for D43
 G = < a,b | a43=b2=1, bab=a-1 >

C1
43C2
C43
D43

Character table of D43

 class 1243A43B43C43D43E43F43G43H43I43J43K43L43M43N43O43P43Q43R43S43T43U
 size 143222222222222222222222
ρ111111111111111111111111    trivial
ρ21-1111111111111111111111    linear of order 2
ρ320ζ4341+ζ432ζ4340+ζ433ζ4339+ζ434ζ4338+ζ435ζ4337+ζ436ζ4336+ζ437ζ4335+ζ438ζ4334+ζ439ζ4333+ζ4310ζ4332+ζ4311ζ4331+ζ4312ζ4330+ζ4313ζ4329+ζ4314ζ4328+ζ4315ζ4327+ζ4316ζ4326+ζ4317ζ4325+ζ4318ζ4324+ζ4319ζ4323+ζ4320ζ4322+ζ4321ζ4342+ζ43    orthogonal faithful
ρ420ζ4333+ζ4310ζ4328+ζ4315ζ4323+ζ4320ζ4325+ζ4318ζ4330+ζ4313ζ4335+ζ438ζ4340+ζ433ζ4341+ζ432ζ4336+ζ437ζ4331+ζ4312ζ4326+ζ4317ζ4322+ζ4321ζ4327+ζ4316ζ4332+ζ4311ζ4337+ζ436ζ4342+ζ43ζ4339+ζ434ζ4334+ζ439ζ4329+ζ4314ζ4324+ζ4319ζ4338+ζ435    orthogonal faithful
ρ520ζ4327+ζ4316ζ4324+ζ4319ζ4332+ζ4311ζ4340+ζ433ζ4338+ζ435ζ4330+ζ4313ζ4322+ζ4321ζ4329+ζ4314ζ4337+ζ436ζ4341+ζ432ζ4333+ζ4310ζ4325+ζ4318ζ4326+ζ4317ζ4334+ζ439ζ4342+ζ43ζ4336+ζ437ζ4328+ζ4315ζ4323+ζ4320ζ4331+ζ4312ζ4339+ζ434ζ4335+ζ438    orthogonal faithful
ρ620ζ4329+ζ4314ζ4322+ζ4321ζ4328+ζ4315ζ4335+ζ438ζ4342+ζ43ζ4337+ζ436ζ4330+ζ4313ζ4323+ζ4320ζ4327+ζ4316ζ4334+ζ439ζ4341+ζ432ζ4338+ζ435ζ4331+ζ4312ζ4324+ζ4319ζ4326+ζ4317ζ4333+ζ4310ζ4340+ζ433ζ4339+ζ434ζ4332+ζ4311ζ4325+ζ4318ζ4336+ζ437    orthogonal faithful
ρ720ζ4337+ζ436ζ4334+ζ439ζ4331+ζ4312ζ4328+ζ4315ζ4325+ζ4318ζ4322+ζ4321ζ4324+ζ4319ζ4327+ζ4316ζ4330+ζ4313ζ4333+ζ4310ζ4336+ζ437ζ4339+ζ434ζ4342+ζ43ζ4341+ζ432ζ4338+ζ435ζ4335+ζ438ζ4332+ζ4311ζ4329+ζ4314ζ4326+ζ4317ζ4323+ζ4320ζ4340+ζ433    orthogonal faithful
ρ820ζ4331+ζ4312ζ4325+ζ4318ζ4324+ζ4319ζ4330+ζ4313ζ4336+ζ437ζ4342+ζ43ζ4338+ζ435ζ4332+ζ4311ζ4326+ζ4317ζ4323+ζ4320ζ4329+ζ4314ζ4335+ζ438ζ4341+ζ432ζ4339+ζ434ζ4333+ζ4310ζ4327+ζ4316ζ4322+ζ4321ζ4328+ζ4315ζ4334+ζ439ζ4340+ζ433ζ4337+ζ436    orthogonal faithful
ρ920ζ4324+ζ4319ζ4336+ζ437ζ4338+ζ435ζ4326+ζ4317ζ4329+ζ4314ζ4341+ζ432ζ4333+ζ4310ζ4322+ζ4321ζ4334+ζ439ζ4340+ζ433ζ4328+ζ4315ζ4327+ζ4316ζ4339+ζ434ζ4335+ζ438ζ4323+ζ4320ζ4332+ζ4311ζ4342+ζ43ζ4330+ζ4313ζ4325+ζ4318ζ4337+ζ436ζ4331+ζ4312    orthogonal faithful
ρ1020ζ4322+ζ4321ζ4333+ζ4310ζ4342+ζ43ζ4331+ζ4312ζ4323+ζ4320ζ4334+ζ439ζ4341+ζ432ζ4330+ζ4313ζ4324+ζ4319ζ4335+ζ438ζ4340+ζ433ζ4329+ζ4314ζ4325+ζ4318ζ4336+ζ437ζ4339+ζ434ζ4328+ζ4315ζ4326+ζ4317ζ4337+ζ436ζ4338+ζ435ζ4327+ζ4316ζ4332+ζ4311    orthogonal faithful
ρ1120ζ4323+ζ4320ζ4330+ζ4313ζ4340+ζ433ζ4336+ζ437ζ4326+ζ4317ζ4327+ζ4316ζ4337+ζ436ζ4339+ζ434ζ4329+ζ4314ζ4324+ζ4319ζ4334+ζ439ζ4342+ζ43ζ4332+ζ4311ζ4322+ζ4321ζ4331+ζ4312ζ4341+ζ432ζ4335+ζ438ζ4325+ζ4318ζ4328+ζ4315ζ4338+ζ435ζ4333+ζ4310    orthogonal faithful
ρ1220ζ4342+ζ43ζ4323+ζ4320ζ4341+ζ432ζ4324+ζ4319ζ4340+ζ433ζ4325+ζ4318ζ4339+ζ434ζ4326+ζ4317ζ4338+ζ435ζ4327+ζ4316ζ4337+ζ436ζ4328+ζ4315ζ4336+ζ437ζ4329+ζ4314ζ4335+ζ438ζ4330+ζ4313ζ4334+ζ439ζ4331+ζ4312ζ4333+ζ4310ζ4332+ζ4311ζ4322+ζ4321    orthogonal faithful
ρ1320ζ4338+ζ435ζ4329+ζ4314ζ4333+ζ4310ζ4334+ζ439ζ4328+ζ4315ζ4339+ζ434ζ4323+ζ4320ζ4342+ζ43ζ4325+ζ4318ζ4337+ζ436ζ4330+ζ4313ζ4332+ζ4311ζ4335+ζ438ζ4327+ζ4316ζ4340+ζ433ζ4322+ζ4321ζ4341+ζ432ζ4326+ζ4317ζ4336+ζ437ζ4331+ζ4312ζ4324+ζ4319    orthogonal faithful
ρ1420ζ4335+ζ438ζ4331+ζ4312ζ4327+ζ4316ζ4323+ζ4320ζ4324+ζ4319ζ4328+ζ4315ζ4332+ζ4311ζ4336+ζ437ζ4340+ζ433ζ4342+ζ43ζ4338+ζ435ζ4334+ζ439ζ4330+ζ4313ζ4326+ζ4317ζ4322+ζ4321ζ4325+ζ4318ζ4329+ζ4314ζ4333+ζ4310ζ4337+ζ436ζ4341+ζ432ζ4339+ζ434    orthogonal faithful
ρ1520ζ4332+ζ4311ζ4338+ζ435ζ4322+ζ4321ζ4337+ζ436ζ4333+ζ4310ζ4326+ζ4317ζ4342+ζ43ζ4328+ζ4315ζ4331+ζ4312ζ4339+ζ434ζ4323+ζ4320ζ4336+ζ437ζ4334+ζ439ζ4325+ζ4318ζ4341+ζ432ζ4329+ζ4314ζ4330+ζ4313ζ4340+ζ433ζ4324+ζ4319ζ4335+ζ438ζ4327+ζ4316    orthogonal faithful
ρ1620ζ4328+ζ4315ζ4342+ζ43ζ4330+ζ4313ζ4327+ζ4316ζ4341+ζ432ζ4331+ζ4312ζ4326+ζ4317ζ4340+ζ433ζ4332+ζ4311ζ4325+ζ4318ζ4339+ζ434ζ4333+ζ4310ζ4324+ζ4319ζ4338+ζ435ζ4334+ζ439ζ4323+ζ4320ζ4337+ζ436ζ4335+ζ438ζ4322+ζ4321ζ4336+ζ437ζ4329+ζ4314    orthogonal faithful
ρ1720ζ4336+ζ437ζ4332+ζ4311ζ4329+ζ4314ζ4339+ζ434ζ4322+ζ4321ζ4340+ζ433ζ4328+ζ4315ζ4333+ζ4310ζ4335+ζ438ζ4326+ζ4317ζ4342+ζ43ζ4324+ζ4319ζ4337+ζ436ζ4331+ζ4312ζ4330+ζ4313ζ4338+ζ435ζ4323+ζ4320ζ4341+ζ432ζ4327+ζ4316ζ4334+ζ439ζ4325+ζ4318    orthogonal faithful
ρ1820ζ4325+ζ4318ζ4327+ζ4316ζ4336+ζ437ζ4341+ζ432ζ4332+ζ4311ζ4323+ζ4320ζ4329+ζ4314ζ4338+ζ435ζ4339+ζ434ζ4330+ζ4313ζ4322+ζ4321ζ4331+ζ4312ζ4340+ζ433ζ4337+ζ436ζ4328+ζ4315ζ4324+ζ4319ζ4333+ζ4310ζ4342+ζ43ζ4335+ζ438ζ4326+ζ4317ζ4334+ζ439    orthogonal faithful
ρ1920ζ4330+ζ4313ζ4341+ζ432ζ4326+ζ4317ζ4332+ζ4311ζ4339+ζ434ζ4324+ζ4319ζ4334+ζ439ζ4337+ζ436ζ4322+ζ4321ζ4336+ζ437ζ4335+ζ438ζ4323+ζ4320ζ4338+ζ435ζ4333+ζ4310ζ4325+ζ4318ζ4340+ζ433ζ4331+ζ4312ζ4327+ζ4316ζ4342+ζ43ζ4329+ζ4314ζ4328+ζ4315    orthogonal faithful
ρ2020ζ4326+ζ4317ζ4339+ζ434ζ4334+ζ439ζ4322+ζ4321ζ4335+ζ438ζ4338+ζ435ζ4325+ζ4318ζ4331+ζ4312ζ4342+ζ43ζ4329+ζ4314ζ4327+ζ4316ζ4340+ζ433ζ4333+ζ4310ζ4323+ζ4320ζ4336+ζ437ζ4337+ζ436ζ4324+ζ4319ζ4332+ζ4311ζ4341+ζ432ζ4328+ζ4315ζ4330+ζ4313    orthogonal faithful
ρ2120ζ4334+ζ439ζ4335+ζ438ζ4325+ζ4318ζ4342+ζ43ζ4327+ζ4316ζ4333+ζ4310ζ4336+ζ437ζ4324+ζ4319ζ4341+ζ432ζ4328+ζ4315ζ4332+ζ4311ζ4337+ζ436ζ4323+ζ4320ζ4340+ζ433ζ4329+ζ4314ζ4331+ζ4312ζ4338+ζ435ζ4322+ζ4321ζ4339+ζ434ζ4330+ζ4313ζ4326+ζ4317    orthogonal faithful
ρ2220ζ4340+ζ433ζ4326+ζ4317ζ4337+ζ436ζ4329+ζ4314ζ4334+ζ439ζ4332+ζ4311ζ4331+ζ4312ζ4335+ζ438ζ4328+ζ4315ζ4338+ζ435ζ4325+ζ4318ζ4341+ζ432ζ4322+ζ4321ζ4342+ζ43ζ4324+ζ4319ζ4339+ζ434ζ4327+ζ4316ζ4336+ζ437ζ4330+ζ4313ζ4333+ζ4310ζ4323+ζ4320    orthogonal faithful
ρ2320ζ4339+ζ434ζ4337+ζ436ζ4335+ζ438ζ4333+ζ4310ζ4331+ζ4312ζ4329+ζ4314ζ4327+ζ4316ζ4325+ζ4318ζ4323+ζ4320ζ4322+ζ4321ζ4324+ζ4319ζ4326+ζ4317ζ4328+ζ4315ζ4330+ζ4313ζ4332+ζ4311ζ4334+ζ439ζ4336+ζ437ζ4338+ζ435ζ4340+ζ433ζ4342+ζ43ζ4341+ζ432    orthogonal faithful

Smallest permutation representation of D43
►On 43 points: primitive
Generators in S43
(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)

(1 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)
 
G:=sub<Sym(43)| (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)>;
 
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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23) );
 
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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23)]])
 

D43 is a maximal subgroup of   C43⋊C6  D129  D215
D43 is a maximal quotient of   Dic43  D129  D215

Matrix representation of D43 ►in GL2(𝔽173) generated by

39172
10
,
39172
136134
G:=sub<GL(2,GF(173))| [39,1,172,0],[39,136,172,134] >;
 

D43 in GAP, Magma, Sage, TeX 

D_{43}
 
% in TeX
 
G:=Group("D43");
 
// GroupNames label
 
G:=SmallGroup(86,1);
 
// by ID
 
G=gap.SmallGroup(86,1);
 
# by ID
 
G:=PCGroup([2,-2,-43,337]);
 
// Polycyclic
 
G:=Group<a,b|a^43=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D43 in TeX
Character table of D43 in TeX

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