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G = D40  order 80 = 24·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D40, C5⋊1D8, C8⋊1D5, C40⋊1C2, D20⋊1C2, C4.9D10, C2.4D20, C10.2D4, C20.9C22, sometimes denoted D80 or Dih40 or Dih80, SmallGroup(80,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D40
Chief series
C1 — C5 — C10 — C20 — D20 — D40
Lower central
C5 — C10 — C20 — D40
Upper central
C1 — C2 — C4 — C8

Generators and relations for D40
 G = < a,b | a40=b2=1, bab=a-1 >

C1
C2
20C2
20C2
C5
C4
10C22
10C22
C10
4D5
4D5
C8
5D4
5D4
C20
2D10
2D10
5D8
D20
C40
D20
D40

Character table of D40

 class 12A2B2C45A5B8A8B10A10B20A20B20C20D40A40B40C40D40E40F40G40H
 size 1120202222222222222222222
ρ111111111111111111111111    trivial
ρ211-1-11111111111111111111    linear of order 2
ρ3111-1111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-11111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ52200-2220022-2-2-2-200000000    orthogonal lifted from D4
ρ622002-1+√5/2-1-√5/2-2-2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21-√5/21-√5/21+√5/21+√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ722002-1+√5/2-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ822002-1-√5/2-1+√5/2-2-2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21+√5/21+√5/21-√5/21-√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ922002-1-√5/2-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ102-200022-√2√2-2-20000√2√2-√2-√2-√2-√2√2√2    orthogonal lifted from D8
ρ112-200022√2-√2-2-20000-√2-√2√2√2√2√2-√2-√2    orthogonal lifted from D8
ρ122200-2-1+√5/2-1-√5/200-1+√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/2-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ132200-2-1-√5/2-1+√5/200-1-√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/2ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ142200-2-1-√5/2-1+√5/200-1-√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/2-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ152200-2-1+√5/2-1-√5/200-1+√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/2ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ162-2000-1-√5/2-1+√5/2-√2√21+√5/21-√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53ζ83ζ52-ζ8ζ53-ζ87ζ54+ζ85ζ5ζ83ζ5-ζ8ζ54ζ83ζ53-ζ8ζ52ζ87ζ54-ζ85ζ5-ζ83ζ5+ζ8ζ54    orthogonal faithful
ρ172-2000-1+√5/2-1-√5/2-√2√21-√5/21+√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5ζ87ζ54-ζ85ζ5-ζ83ζ5+ζ8ζ54ζ83ζ5-ζ8ζ54ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52-ζ87ζ54+ζ85ζ5-ζ83ζ52+ζ8ζ53-ζ83ζ53+ζ8ζ52    orthogonal faithful
ρ182-2000-1-√5/2-1+√5/2√2-√21+√5/21-√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52-ζ83ζ53+ζ8ζ52-ζ83ζ5+ζ8ζ54ζ87ζ54-ζ85ζ5-ζ83ζ52+ζ8ζ53ζ83ζ5-ζ8ζ54-ζ87ζ54+ζ85ζ5    orthogonal faithful
ρ192-2000-1+√5/2-1-√5/2√2-√21-√5/21+√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5-ζ87ζ54+ζ85ζ5ζ83ζ5-ζ8ζ54-ζ83ζ5+ζ8ζ54-ζ83ζ52+ζ8ζ53-ζ83ζ53+ζ8ζ52ζ87ζ54-ζ85ζ5ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52    orthogonal faithful
ρ202-2000-1+√5/2-1-√5/2√2-√21-√5/21+√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5ζ83ζ5-ζ8ζ54-ζ87ζ54+ζ85ζ5ζ87ζ54-ζ85ζ5-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53-ζ83ζ5+ζ8ζ54ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53    orthogonal faithful
ρ212-2000-1+√5/2-1-√5/2-√2√21-√5/21+√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5-ζ83ζ5+ζ8ζ54ζ87ζ54-ζ85ζ5-ζ87ζ54+ζ85ζ5ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53ζ83ζ5-ζ8ζ54-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53    orthogonal faithful
ρ222-2000-1-√5/2-1+√5/2-√2√21+√5/21-√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52-ζ83ζ52+ζ8ζ53-ζ83ζ53+ζ8ζ52ζ83ζ53-ζ8ζ52ζ83ζ5-ζ8ζ54-ζ87ζ54+ζ85ζ5ζ83ζ52-ζ8ζ53-ζ83ζ5+ζ8ζ54ζ87ζ54-ζ85ζ5    orthogonal faithful
ρ232-2000-1-√5/2-1+√5/2√2-√21+√5/21-√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53-ζ83ζ52+ζ8ζ53ζ87ζ54-ζ85ζ5-ζ83ζ5+ζ8ζ54-ζ83ζ53+ζ8ζ52-ζ87ζ54+ζ85ζ5ζ83ζ5-ζ8ζ54    orthogonal faithful

Smallest permutation representation of D40
►On 40 points
Generators in S40
(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)

(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)
 
G:=sub<Sym(40)| (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)>;
 
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21) );
 
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21)]])
 

D40 is a maximal subgroup of
 D80  C16⋊D5  C5⋊D16  C5⋊SD32  D40⋊7C2  C8⋊D10  D5×D8  D40⋊C2  Q8.D10  C3⋊D40  D120  D200  C5⋊D40  C52⋊5D8
D40 is a maximal quotient of
 D80  C16⋊D5  Dic40  C40⋊5C4  D20⋊5C4  C3⋊D40  D120  D200  C5⋊D40  C52⋊5D8

Matrix representation of D40 ►in GL2(𝔽41) generated by

53
3823
,
1218
829
G:=sub<GL(2,GF(41))| [5,38,3,23],[12,8,18,29] >;
 

D40 in GAP, Magma, Sage, TeX 

D_{40}
 
% in TeX
 
G:=Group("D40");
 
// GroupNames label
 
G:=SmallGroup(80,7);
 
// by ID
 
G=gap.SmallGroup(80,7);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-5,61,66,182,42,1604]);
 
// Polycyclic
 
G:=Group<a,b|a^40=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D40 in TeX
Character table of D40 in TeX

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