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G = D40order 80 = 24·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D40, C51D8, C81D5, C401C2, D201C2, 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

C1C20 — D40
C1C5C10C20D20 — D40
C5C10C20 — D40
C1C2C4C8

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

20C2
20C2
10C22
10C22
4D5
4D5
5D4
5D4
2D10
2D10
5D8

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
ρ102-200022-22-2-2000022-2-2-2-222    orthogonal lifted from D8
ρ112-2000222-2-2-20000-2-22222-2-2    orthogonal lifted from D8
ρ122200-2-1+5/2-1-5/200-1+5/2-1-5/21-5/21+5/21+5/21-5/243ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    orthogonal lifted from D20
ρ132200-2-1-5/2-1+5/200-1-5/2-1+5/21+5/21-5/21-5/21+5/2ζ4ζ534ζ524ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    orthogonal lifted from D20
ρ142200-2-1-5/2-1+5/200-1-5/2-1+5/21+5/21-5/21-5/21+5/24ζ534ζ52ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    orthogonal lifted from D20
ρ152200-2-1+5/2-1-5/200-1+5/2-1-5/21-5/21+5/21+5/21-5/2ζ43ζ5443ζ543ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ162-2000-1-5/2-1+5/2-221+5/21-5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ5283ζ538ζ5283ζ528ζ53ζ83ζ528ζ5387ζ5485ζ5ζ83ζ58ζ54ζ83ζ538ζ52ζ87ζ5485ζ583ζ58ζ54    orthogonal faithful
ρ172-2000-1+5/2-1-5/2-221-5/21+5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ5ζ87ζ5485ζ583ζ58ζ54ζ83ζ58ζ54ζ83ζ528ζ53ζ83ζ538ζ5287ζ5485ζ583ζ528ζ5383ζ538ζ52    orthogonal faithful
ρ182-2000-1-5/2-1+5/22-21+5/21-5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ52ζ83ζ528ζ53ζ83ζ538ζ5283ζ538ζ5283ζ58ζ54ζ87ζ5485ζ583ζ528ζ53ζ83ζ58ζ5487ζ5485ζ5    orthogonal faithful
ρ192-2000-1+5/2-1-5/22-21-5/21+5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ587ζ5485ζ5ζ83ζ58ζ5483ζ58ζ5483ζ528ζ5383ζ538ζ52ζ87ζ5485ζ5ζ83ζ528ζ53ζ83ζ538ζ52    orthogonal faithful
ρ202-2000-1+5/2-1-5/22-21-5/21+5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ5ζ83ζ58ζ5487ζ5485ζ5ζ87ζ5485ζ583ζ538ζ5283ζ528ζ5383ζ58ζ54ζ83ζ538ζ52ζ83ζ528ζ53    orthogonal faithful
ρ212-2000-1+5/2-1-5/2-221-5/21+5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ583ζ58ζ54ζ87ζ5485ζ587ζ5485ζ5ζ83ζ538ζ52ζ83ζ528ζ53ζ83ζ58ζ5483ζ538ζ5283ζ528ζ53    orthogonal faithful
ρ222-2000-1-5/2-1+5/2-221+5/21-5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ5283ζ528ζ5383ζ538ζ52ζ83ζ538ζ52ζ83ζ58ζ5487ζ5485ζ5ζ83ζ528ζ5383ζ58ζ54ζ87ζ5485ζ5    orthogonal faithful
ρ232-2000-1-5/2-1+5/22-21+5/21-5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ83ζ538ζ52ζ83ζ528ζ5383ζ528ζ53ζ87ζ5485ζ583ζ58ζ5483ζ538ζ5287ζ5485ζ5ζ83ζ58ζ54    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  D407C2  C8⋊D10  D5×D8  D40⋊C2  Q8.D10  C3⋊D40  D120  D200  C5⋊D40  C525D8
D40 is a maximal quotient of
D80  C16⋊D5  Dic40  C405C4  D205C4  C3⋊D40  D120  D200  C5⋊D40  C525D8

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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