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G = D41  order 82 = 2·41

Dihedral group

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

Aliases: D41, C41⋊C2, sometimes denoted D82 or Dih41 or Dih82, SmallGroup(82,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C41 — D41
Chief series
C1 — C41 — D41
Lower central
C41 — D41
Upper central
C1

Generators and relations for D41
 G = < a,b | a41=b2=1, bab=a-1 >

C1
41C2
C41
D41

Character table of D41

 class 1241A41B41C41D41E41F41G41H41I41J41K41L41M41N41O41P41Q41R41S41T
 size 14122222222222222222222
ρ11111111111111111111111    trivial
ρ21-111111111111111111111    linear of order 2
ρ320ζ4133+ζ418ζ4129+ζ4112ζ4125+ζ4116ζ4121+ζ4120ζ4124+ζ4117ζ4128+ζ4113ζ4132+ζ419ζ4136+ζ415ζ4140+ζ41ζ4138+ζ413ζ4134+ζ417ζ4130+ζ4111ζ4126+ζ4115ζ4122+ζ4119ζ4123+ζ4118ζ4127+ζ4114ζ4131+ζ4110ζ4135+ζ416ζ4139+ζ412ζ4137+ζ414    orthogonal faithful
ρ420ζ4131+ζ4110ζ4126+ζ4115ζ4121+ζ4120ζ4125+ζ4116ζ4130+ζ4111ζ4135+ζ416ζ4140+ζ41ζ4137+ζ414ζ4132+ζ419ζ4127+ζ4114ζ4122+ζ4119ζ4124+ζ4117ζ4129+ζ4112ζ4134+ζ417ζ4139+ζ412ζ4138+ζ413ζ4133+ζ418ζ4128+ζ4113ζ4123+ζ4118ζ4136+ζ415    orthogonal faithful
ρ520ζ4128+ζ4113ζ4140+ζ41ζ4126+ζ4115ζ4129+ζ4112ζ4139+ζ412ζ4125+ζ4116ζ4130+ζ4111ζ4138+ζ413ζ4124+ζ4117ζ4131+ζ4110ζ4137+ζ414ζ4123+ζ4118ζ4132+ζ419ζ4136+ζ415ζ4122+ζ4119ζ4133+ζ418ζ4135+ζ416ζ4121+ζ4120ζ4134+ζ417ζ4127+ζ4114    orthogonal faithful
ρ620ζ4136+ζ415ζ4128+ζ4113ζ4131+ζ4110ζ4133+ζ418ζ4126+ζ4115ζ4138+ζ413ζ4121+ζ4120ζ4139+ζ412ζ4125+ζ4116ζ4134+ζ417ζ4130+ζ4111ζ4129+ζ4112ζ4135+ζ416ζ4124+ζ4117ζ4140+ζ41ζ4122+ζ4119ζ4137+ζ414ζ4127+ζ4114ζ4132+ζ419ζ4123+ζ4118    orthogonal faithful
ρ720ζ4122+ζ4119ζ4133+ζ418ζ4138+ζ413ζ4127+ζ4114ζ4125+ζ4116ζ4136+ζ415ζ4135+ζ416ζ4124+ζ4117ζ4128+ζ4113ζ4139+ζ412ζ4132+ζ419ζ4121+ζ4120ζ4131+ζ4110ζ4140+ζ41ζ4129+ζ4112ζ4123+ζ4118ζ4134+ζ417ζ4137+ζ414ζ4126+ζ4115ζ4130+ζ4111    orthogonal faithful
ρ820ζ4127+ζ4114ζ4121+ζ4120ζ4128+ζ4113ζ4135+ζ416ζ4140+ζ41ζ4133+ζ418ζ4126+ζ4115ζ4122+ζ4119ζ4129+ζ4112ζ4136+ζ415ζ4139+ζ412ζ4132+ζ419ζ4125+ζ4116ζ4123+ζ4118ζ4130+ζ4111ζ4137+ζ414ζ4138+ζ413ζ4131+ζ4110ζ4124+ζ4117ζ4134+ζ417    orthogonal faithful
ρ920ζ4138+ζ413ζ4125+ζ4116ζ4135+ζ416ζ4128+ζ4113ζ4132+ζ419ζ4131+ζ4110ζ4129+ζ4112ζ4134+ζ417ζ4126+ζ4115ζ4137+ζ414ζ4123+ζ4118ζ4140+ζ41ζ4121+ζ4120ζ4139+ζ412ζ4124+ζ4117ζ4136+ζ415ζ4127+ζ4114ζ4133+ζ418ζ4130+ζ4111ζ4122+ζ4119    orthogonal faithful
ρ1020ζ4121+ζ4120ζ4130+ζ4111ζ4140+ζ41ζ4132+ζ419ζ4122+ζ4119ζ4129+ζ4112ζ4139+ζ412ζ4133+ζ418ζ4123+ζ4118ζ4128+ζ4113ζ4138+ζ413ζ4134+ζ417ζ4124+ζ4117ζ4127+ζ4114ζ4137+ζ414ζ4135+ζ416ζ4125+ζ4116ζ4126+ζ4115ζ4136+ζ415ζ4131+ζ4110    orthogonal faithful
ρ1120ζ4140+ζ41ζ4122+ζ4119ζ4139+ζ412ζ4123+ζ4118ζ4138+ζ413ζ4124+ζ4117ζ4137+ζ414ζ4125+ζ4116ζ4136+ζ415ζ4126+ζ4115ζ4135+ζ416ζ4127+ζ4114ζ4134+ζ417ζ4128+ζ4113ζ4133+ζ418ζ4129+ζ4112ζ4132+ζ419ζ4130+ζ4111ζ4131+ζ4110ζ4121+ζ4120    orthogonal faithful
ρ1220ζ4123+ζ4118ζ4127+ζ4114ζ4136+ζ415ζ4137+ζ414ζ4128+ζ4113ζ4122+ζ4119ζ4131+ζ4110ζ4140+ζ41ζ4133+ζ418ζ4124+ζ4117ζ4126+ζ4115ζ4135+ζ416ζ4138+ζ413ζ4129+ζ4112ζ4121+ζ4120ζ4130+ζ4111ζ4139+ζ412ζ4134+ζ417ζ4125+ζ4116ζ4132+ζ419    orthogonal faithful
ρ1320ζ4137+ζ414ζ4135+ζ416ζ4133+ζ418ζ4131+ζ4110ζ4129+ζ4112ζ4127+ζ4114ζ4125+ζ4116ζ4123+ζ4118ζ4121+ζ4120ζ4122+ζ4119ζ4124+ζ4117ζ4126+ζ4115ζ4128+ζ4113ζ4130+ζ4111ζ4132+ζ419ζ4134+ζ417ζ4136+ζ415ζ4138+ζ413ζ4140+ζ41ζ4139+ζ412    orthogonal faithful
ρ1420ζ4139+ζ412ζ4138+ζ413ζ4137+ζ414ζ4136+ζ415ζ4135+ζ416ζ4134+ζ417ζ4133+ζ418ζ4132+ζ419ζ4131+ζ4110ζ4130+ζ4111ζ4129+ζ4112ζ4128+ζ4113ζ4127+ζ4114ζ4126+ζ4115ζ4125+ζ4116ζ4124+ζ4117ζ4123+ζ4118ζ4122+ζ4119ζ4121+ζ4120ζ4140+ζ41    orthogonal faithful
ρ1520ζ4132+ζ419ζ4134+ζ417ζ4123+ζ4118ζ4139+ζ412ζ4127+ζ4114ζ4130+ζ4111ζ4136+ζ415ζ4121+ζ4120ζ4137+ζ414ζ4129+ζ4112ζ4128+ζ4113ζ4138+ζ413ζ4122+ζ4119ζ4135+ζ416ζ4131+ζ4110ζ4126+ζ4115ζ4140+ζ41ζ4124+ζ4117ζ4133+ζ418ζ4125+ζ4116    orthogonal faithful
ρ1620ζ4124+ζ4117ζ4136+ζ415ζ4134+ζ417ζ4122+ζ4119ζ4131+ζ4110ζ4139+ζ412ζ4127+ζ4114ζ4126+ζ4115ζ4138+ζ413ζ4132+ζ419ζ4121+ζ4120ζ4133+ζ418ζ4137+ζ414ζ4125+ζ4116ζ4128+ζ4113ζ4140+ζ41ζ4130+ζ4111ζ4123+ζ4118ζ4135+ζ416ζ4129+ζ4112    orthogonal faithful
ρ1720ζ4135+ζ416ζ4132+ζ419ζ4129+ζ4112ζ4126+ζ4115ζ4123+ζ4118ζ4121+ζ4120ζ4124+ζ4117ζ4127+ζ4114ζ4130+ζ4111ζ4133+ζ418ζ4136+ζ415ζ4139+ζ412ζ4140+ζ41ζ4137+ζ414ζ4134+ζ417ζ4131+ζ4110ζ4128+ζ4113ζ4125+ζ4116ζ4122+ζ4119ζ4138+ζ413    orthogonal faithful
ρ1820ζ4134+ζ417ζ4131+ζ4110ζ4127+ζ4114ζ4138+ζ413ζ4121+ζ4120ζ4137+ζ414ζ4128+ζ4113ζ4130+ζ4111ζ4135+ζ416ζ4123+ζ4118ζ4140+ζ41ζ4125+ζ4116ζ4133+ζ418ζ4132+ζ419ζ4126+ζ4115ζ4139+ζ412ζ4122+ζ4119ζ4136+ζ415ζ4129+ζ4112ζ4124+ζ4117    orthogonal faithful
ρ1920ζ4125+ζ4116ζ4124+ζ4117ζ4132+ζ419ζ4140+ζ41ζ4134+ζ417ζ4126+ζ4115ζ4123+ζ4118ζ4131+ζ4110ζ4139+ζ412ζ4135+ζ416ζ4127+ζ4114ζ4122+ζ4119ζ4130+ζ4111ζ4138+ζ413ζ4136+ζ415ζ4128+ζ4113ζ4121+ζ4120ζ4129+ζ4112ζ4137+ζ414ζ4133+ζ418    orthogonal faithful
ρ2020ζ4126+ζ4115ζ4139+ζ412ζ4130+ζ4111ζ4124+ζ4117ζ4137+ζ414ζ4132+ζ419ζ4122+ζ4119ζ4135+ζ416ζ4134+ζ417ζ4121+ζ4120ζ4133+ζ418ζ4136+ζ415ζ4123+ζ4118ζ4131+ζ4110ζ4138+ζ413ζ4125+ζ4116ζ4129+ζ4112ζ4140+ζ41ζ4127+ζ4114ζ4128+ζ4113    orthogonal faithful
ρ2120ζ4129+ζ4112ζ4123+ζ4118ζ4124+ζ4117ζ4130+ζ4111ζ4136+ζ415ζ4140+ζ41ζ4134+ζ417ζ4128+ζ4113ζ4122+ζ4119ζ4125+ζ4116ζ4131+ζ4110ζ4137+ζ414ζ4139+ζ412ζ4133+ζ418ζ4127+ζ4114ζ4121+ζ4120ζ4126+ζ4115ζ4132+ζ419ζ4138+ζ413ζ4135+ζ416    orthogonal faithful
ρ2220ζ4130+ζ4111ζ4137+ζ414ζ4122+ζ4119ζ4134+ζ417ζ4133+ζ418ζ4123+ζ4118ζ4138+ζ413ζ4129+ζ4112ζ4127+ζ4114ζ4140+ζ41ζ4125+ζ4116ζ4131+ζ4110ζ4136+ζ415ζ4121+ζ4120ζ4135+ζ416ζ4132+ζ419ζ4124+ζ4117ζ4139+ζ412ζ4128+ζ4113ζ4126+ζ4115    orthogonal faithful

Smallest permutation representation of D41
►On 41 points: primitive
Generators in S41
(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)

(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)
 
G:=sub<Sym(41)| (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)>;
 
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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22) );
 
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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22)]])
 

D41 is a maximal subgroup of   C41⋊C4  D123  C41⋊C10  D205
D41 is a maximal quotient of   Dic41  D123  D205

Matrix representation of D41 ►in GL2(𝔽83) generated by

8182
1448
,
4043
1343
G:=sub<GL(2,GF(83))| [81,14,82,48],[40,13,43,43] >;
 

D41 in GAP, Magma, Sage, TeX 

D_{41}
 
% in TeX
 
G:=Group("D41");
 
// GroupNames label
 
G:=SmallGroup(82,1);
 
// by ID
 
G=gap.SmallGroup(82,1);
 
# by ID
 
G:=PCGroup([2,-2,-41,321]);
 
// Polycyclic
 
G:=Group<a,b|a^41=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D41 in TeX
Character table of D41 in TeX

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