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G = D41order 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

C1C41 — D41
C1C41 — D41
C41 — D41
C1

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

41C2

Character table of D41

 class 1241A41B41C41D41E41F41G41H41I41J41K41L41M41N41O41P41Q41R41S41T
 size 14122222222222222222222
ρ11111111111111111111111    trivial
ρ21-111111111111111111111    linear of order 2
ρ320ζ4133418ζ41294112ζ41254116ζ41214120ζ41244117ζ41284113ζ4132419ζ4136415ζ414041ζ4138413ζ4134417ζ41304111ζ41264115ζ41224119ζ41234118ζ41274114ζ41314110ζ4135416ζ4139412ζ4137414    orthogonal faithful
ρ420ζ41314110ζ41264115ζ41214120ζ41254116ζ41304111ζ4135416ζ414041ζ4137414ζ4132419ζ41274114ζ41224119ζ41244117ζ41294112ζ4134417ζ4139412ζ4138413ζ4133418ζ41284113ζ41234118ζ4136415    orthogonal faithful
ρ520ζ41284113ζ414041ζ41264115ζ41294112ζ4139412ζ41254116ζ41304111ζ4138413ζ41244117ζ41314110ζ4137414ζ41234118ζ4132419ζ4136415ζ41224119ζ4133418ζ4135416ζ41214120ζ4134417ζ41274114    orthogonal faithful
ρ620ζ4136415ζ41284113ζ41314110ζ4133418ζ41264115ζ4138413ζ41214120ζ4139412ζ41254116ζ4134417ζ41304111ζ41294112ζ4135416ζ41244117ζ414041ζ41224119ζ4137414ζ41274114ζ4132419ζ41234118    orthogonal faithful
ρ720ζ41224119ζ4133418ζ4138413ζ41274114ζ41254116ζ4136415ζ4135416ζ41244117ζ41284113ζ4139412ζ4132419ζ41214120ζ41314110ζ414041ζ41294112ζ41234118ζ4134417ζ4137414ζ41264115ζ41304111    orthogonal faithful
ρ820ζ41274114ζ41214120ζ41284113ζ4135416ζ414041ζ4133418ζ41264115ζ41224119ζ41294112ζ4136415ζ4139412ζ4132419ζ41254116ζ41234118ζ41304111ζ4137414ζ4138413ζ41314110ζ41244117ζ4134417    orthogonal faithful
ρ920ζ4138413ζ41254116ζ4135416ζ41284113ζ4132419ζ41314110ζ41294112ζ4134417ζ41264115ζ4137414ζ41234118ζ414041ζ41214120ζ4139412ζ41244117ζ4136415ζ41274114ζ4133418ζ41304111ζ41224119    orthogonal faithful
ρ1020ζ41214120ζ41304111ζ414041ζ4132419ζ41224119ζ41294112ζ4139412ζ4133418ζ41234118ζ41284113ζ4138413ζ4134417ζ41244117ζ41274114ζ4137414ζ4135416ζ41254116ζ41264115ζ4136415ζ41314110    orthogonal faithful
ρ1120ζ414041ζ41224119ζ4139412ζ41234118ζ4138413ζ41244117ζ4137414ζ41254116ζ4136415ζ41264115ζ4135416ζ41274114ζ4134417ζ41284113ζ4133418ζ41294112ζ4132419ζ41304111ζ41314110ζ41214120    orthogonal faithful
ρ1220ζ41234118ζ41274114ζ4136415ζ4137414ζ41284113ζ41224119ζ41314110ζ414041ζ4133418ζ41244117ζ41264115ζ4135416ζ4138413ζ41294112ζ41214120ζ41304111ζ4139412ζ4134417ζ41254116ζ4132419    orthogonal faithful
ρ1320ζ4137414ζ4135416ζ4133418ζ41314110ζ41294112ζ41274114ζ41254116ζ41234118ζ41214120ζ41224119ζ41244117ζ41264115ζ41284113ζ41304111ζ4132419ζ4134417ζ4136415ζ4138413ζ414041ζ4139412    orthogonal faithful
ρ1420ζ4139412ζ4138413ζ4137414ζ4136415ζ4135416ζ4134417ζ4133418ζ4132419ζ41314110ζ41304111ζ41294112ζ41284113ζ41274114ζ41264115ζ41254116ζ41244117ζ41234118ζ41224119ζ41214120ζ414041    orthogonal faithful
ρ1520ζ4132419ζ4134417ζ41234118ζ4139412ζ41274114ζ41304111ζ4136415ζ41214120ζ4137414ζ41294112ζ41284113ζ4138413ζ41224119ζ4135416ζ41314110ζ41264115ζ414041ζ41244117ζ4133418ζ41254116    orthogonal faithful
ρ1620ζ41244117ζ4136415ζ4134417ζ41224119ζ41314110ζ4139412ζ41274114ζ41264115ζ4138413ζ4132419ζ41214120ζ4133418ζ4137414ζ41254116ζ41284113ζ414041ζ41304111ζ41234118ζ4135416ζ41294112    orthogonal faithful
ρ1720ζ4135416ζ4132419ζ41294112ζ41264115ζ41234118ζ41214120ζ41244117ζ41274114ζ41304111ζ4133418ζ4136415ζ4139412ζ414041ζ4137414ζ4134417ζ41314110ζ41284113ζ41254116ζ41224119ζ4138413    orthogonal faithful
ρ1820ζ4134417ζ41314110ζ41274114ζ4138413ζ41214120ζ4137414ζ41284113ζ41304111ζ4135416ζ41234118ζ414041ζ41254116ζ4133418ζ4132419ζ41264115ζ4139412ζ41224119ζ4136415ζ41294112ζ41244117    orthogonal faithful
ρ1920ζ41254116ζ41244117ζ4132419ζ414041ζ4134417ζ41264115ζ41234118ζ41314110ζ4139412ζ4135416ζ41274114ζ41224119ζ41304111ζ4138413ζ4136415ζ41284113ζ41214120ζ41294112ζ4137414ζ4133418    orthogonal faithful
ρ2020ζ41264115ζ4139412ζ41304111ζ41244117ζ4137414ζ4132419ζ41224119ζ4135416ζ4134417ζ41214120ζ4133418ζ4136415ζ41234118ζ41314110ζ4138413ζ41254116ζ41294112ζ414041ζ41274114ζ41284113    orthogonal faithful
ρ2120ζ41294112ζ41234118ζ41244117ζ41304111ζ4136415ζ414041ζ4134417ζ41284113ζ41224119ζ41254116ζ41314110ζ4137414ζ4139412ζ4133418ζ41274114ζ41214120ζ41264115ζ4132419ζ4138413ζ4135416    orthogonal faithful
ρ2220ζ41304111ζ4137414ζ41224119ζ4134417ζ4133418ζ41234118ζ4138413ζ41294112ζ41274114ζ414041ζ41254116ζ41314110ζ4136415ζ41214120ζ4135416ζ4132419ζ41244117ζ4139412ζ41284113ζ41264115    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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