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G = D50order 100 = 22·52

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D50, C2×D25, C50⋊C2, C25⋊C22, C5.D10, C10.2D5, sometimes denoted D100 or Dih50 or Dih100, SmallGroup(100,4)

Series: Derived Chief Lower central Upper central

C1C25 — D50
C1C5C25D25 — D50
C25 — D50
C1C2

Generators and relations for D50
 G = < a,b | a50=b2=1, bab=a-1 >

25C2
25C2
25C22
5D5
5D5
5D10

Character table of D50

 class 12A2B2C5A5B10A10B25A25B25C25D25E25F25G25H25I25J50A50B50C50D50E50F50G50H50I50J
 size 112525222222222222222222222222
ρ11111111111111111111111111111    trivial
ρ21-1-1111-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111111111111111111111    linear of order 2
ρ41-11-111-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ522002222-1-5/2-1+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+5/2-1+5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ62-20022-2-2-1+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+5/21-5/21-5/21-5/21+5/21+5/21+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ72-20022-2-2-1-5/2-1+5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1-5/21+5/21+5/21+5/21-5/21-5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ822002222-1+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+5/2-1-5/2-1-5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ92-200-1-5/2-1+5/21-5/21+5/2ζ2516259ζ25132512ζ2518257ζ2523252ζ2522253ζ25142511ζ2519256ζ252425ζ2517258ζ252125425142511251925625242525172582513251225182572523252252225325212542516259    orthogonal faithful
ρ102-200-1+5/2-1-5/21+5/21-5/2ζ2517258ζ2519256ζ2516259ζ252425ζ25142511ζ2518257ζ2522253ζ25132512ζ2521254ζ252325225182572522253251325122521254251925625162592524252514251125232522517258    orthogonal faithful
ρ112200-1-5/2-1+5/2-1+5/2-1-5/2ζ2516259ζ25132512ζ2518257ζ2523252ζ2522253ζ25142511ζ2519256ζ252425ζ2517258ζ2521254ζ25142511ζ2519256ζ252425ζ2517258ζ25132512ζ2518257ζ2523252ζ2522253ζ2521254ζ2516259    orthogonal lifted from D25
ρ122-200-1+5/2-1-5/21+5/21-5/2ζ2523252ζ25142511ζ2521254ζ2519256ζ2516259ζ2517258ζ2518257ζ2522253ζ252425ζ2513251225172582518257252225325242525142511252125425192562516259251325122523252    orthogonal faithful
ρ132-200-1-5/2-1+5/21-5/21+5/2ζ2521254ζ2522253ζ2517258ζ25132512ζ2518257ζ2516259ζ25142511ζ2519256ζ2523252ζ25242525162592514251125192562523252252225325172582513251225182572524252521254    orthogonal faithful
ρ142200-1+5/2-1-5/2-1-5/2-1+5/2ζ2522253ζ2521254ζ2519256ζ2516259ζ252425ζ25132512ζ2523252ζ2517258ζ25142511ζ2518257ζ25132512ζ2523252ζ2517258ζ25142511ζ2521254ζ2519256ζ2516259ζ252425ζ2518257ζ2522253    orthogonal lifted from D25
ρ152200-1-5/2-1+5/2-1+5/2-1-5/2ζ25142511ζ2523252ζ2522253ζ2517258ζ25132512ζ2519256ζ252425ζ2521254ζ2518257ζ2516259ζ2519256ζ252425ζ2521254ζ2518257ζ2523252ζ2522253ζ2517258ζ25132512ζ2516259ζ25142511    orthogonal lifted from D25
ρ162200-1-5/2-1+5/2-1+5/2-1-5/2ζ2521254ζ2522253ζ2517258ζ25132512ζ2518257ζ2516259ζ25142511ζ2519256ζ2523252ζ252425ζ2516259ζ25142511ζ2519256ζ2523252ζ2522253ζ2517258ζ25132512ζ2518257ζ252425ζ2521254    orthogonal lifted from D25
ρ172-200-1-5/2-1+5/21-5/21+5/2ζ252425ζ2518257ζ2523252ζ2522253ζ2517258ζ2521254ζ2516259ζ25142511ζ25132512ζ251925625212542516259251425112513251225182572523252252225325172582519256252425    orthogonal faithful
ρ182200-1+5/2-1-5/2-1-5/2-1+5/2ζ2518257ζ252425ζ25142511ζ2521254ζ2519256ζ2522253ζ25132512ζ2523252ζ2516259ζ2517258ζ2522253ζ25132512ζ2523252ζ2516259ζ252425ζ25142511ζ2521254ζ2519256ζ2517258ζ2518257    orthogonal lifted from D25
ρ192-200-1-5/2-1+5/21-5/21+5/2ζ25142511ζ2523252ζ2522253ζ2517258ζ25132512ζ2519256ζ252425ζ2521254ζ2518257ζ251625925192562524252521254251825725232522522253251725825132512251625925142511    orthogonal faithful
ρ202-200-1+5/2-1-5/21+5/21-5/2ζ2518257ζ252425ζ25142511ζ2521254ζ2519256ζ2522253ζ25132512ζ2523252ζ2516259ζ251725825222532513251225232522516259252425251425112521254251925625172582518257    orthogonal faithful
ρ212200-1+5/2-1-5/2-1-5/2-1+5/2ζ2517258ζ2519256ζ2516259ζ252425ζ25142511ζ2518257ζ2522253ζ25132512ζ2521254ζ2523252ζ2518257ζ2522253ζ25132512ζ2521254ζ2519256ζ2516259ζ252425ζ25142511ζ2523252ζ2517258    orthogonal lifted from D25
ρ222-200-1+5/2-1-5/21+5/21-5/2ζ2522253ζ2521254ζ2519256ζ2516259ζ252425ζ25132512ζ2523252ζ2517258ζ25142511ζ251825725132512252325225172582514251125212542519256251625925242525182572522253    orthogonal faithful
ρ232200-1-5/2-1+5/2-1+5/2-1-5/2ζ2519256ζ2517258ζ25132512ζ2518257ζ2523252ζ252425ζ2521254ζ2516259ζ2522253ζ25142511ζ252425ζ2521254ζ2516259ζ2522253ζ2517258ζ25132512ζ2518257ζ2523252ζ25142511ζ2519256    orthogonal lifted from D25
ρ242200-1+5/2-1-5/2-1-5/2-1+5/2ζ2523252ζ25142511ζ2521254ζ2519256ζ2516259ζ2517258ζ2518257ζ2522253ζ252425ζ25132512ζ2517258ζ2518257ζ2522253ζ252425ζ25142511ζ2521254ζ2519256ζ2516259ζ25132512ζ2523252    orthogonal lifted from D25
ρ252-200-1+5/2-1-5/21+5/21-5/2ζ25132512ζ2516259ζ252425ζ25142511ζ2521254ζ2523252ζ2517258ζ2518257ζ2519256ζ252225325232522517258251825725192562516259252425251425112521254252225325132512    orthogonal faithful
ρ262200-1-5/2-1+5/2-1+5/2-1-5/2ζ252425ζ2518257ζ2523252ζ2522253ζ2517258ζ2521254ζ2516259ζ25142511ζ25132512ζ2519256ζ2521254ζ2516259ζ25142511ζ25132512ζ2518257ζ2523252ζ2522253ζ2517258ζ2519256ζ252425    orthogonal lifted from D25
ρ272200-1+5/2-1-5/2-1-5/2-1+5/2ζ25132512ζ2516259ζ252425ζ25142511ζ2521254ζ2523252ζ2517258ζ2518257ζ2519256ζ2522253ζ2523252ζ2517258ζ2518257ζ2519256ζ2516259ζ252425ζ25142511ζ2521254ζ2522253ζ25132512    orthogonal lifted from D25
ρ282-200-1-5/2-1+5/21-5/21+5/2ζ2519256ζ2517258ζ25132512ζ2518257ζ2523252ζ252425ζ2521254ζ2516259ζ2522253ζ2514251125242525212542516259252225325172582513251225182572523252251425112519256    orthogonal faithful

Smallest permutation representation of D50
On 50 points
Generators in S50
(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 44 45 46 47 48 49 50)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)

G:=sub<Sym(50)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26)])

D50 is a maximal subgroup of   D100  C25⋊D4
D50 is a maximal quotient of   Dic50  D100  C25⋊D4

Matrix representation of D50 in GL2(𝔽101) generated by

6451
8952
,
46
4897
G:=sub<GL(2,GF(101))| [64,89,51,52],[4,48,6,97] >;

D50 in GAP, Magma, Sage, TeX

D_{50}
% in TeX

G:=Group("D50");
// GroupNames label

G:=SmallGroup(100,4);
// by ID

G=gap.SmallGroup(100,4);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,434,250,1283]);
// Polycyclic

G:=Group<a,b|a^50=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D50 in TeX
Character table of D50 in TeX

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