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G = D50  order 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

Derived series
C1 — C25 — D50
Chief series
C1 — C5 — C25 — D25 — D50
Lower central
C25 — D50
Upper central
C1 — C2

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

C1
C2
25C2
25C2
C5
25C22
C10
5D5
5D5
C25
5D10
C50
D25
D25
D50

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+√5/2ζ2516+ζ259ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2521+ζ254-ζ2514-ζ2511-ζ2519-ζ256-ζ2524-ζ25-ζ2517-ζ258-ζ2513-ζ2512-ζ2518-ζ257-ζ2523-ζ252-ζ2522-ζ253-ζ2521-ζ254-ζ2516-ζ259    orthogonal faithful
ρ102-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2517+ζ258ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2523+ζ252-ζ2518-ζ257-ζ2522-ζ253-ζ2513-ζ2512-ζ2521-ζ254-ζ2519-ζ256-ζ2516-ζ259-ζ2524-ζ25-ζ2514-ζ2511-ζ2523-ζ252-ζ2517-ζ258    orthogonal faithful
ρ112200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2516+ζ259ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2521+ζ254ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2521+ζ254ζ2516+ζ259    orthogonal lifted from D25
ρ122-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2523+ζ252ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2513+ζ2512-ζ2517-ζ258-ζ2518-ζ257-ζ2522-ζ253-ζ2524-ζ25-ζ2514-ζ2511-ζ2521-ζ254-ζ2519-ζ256-ζ2516-ζ259-ζ2513-ζ2512-ζ2523-ζ252    orthogonal faithful
ρ132-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2521+ζ254ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2524+ζ25-ζ2516-ζ259-ζ2514-ζ2511-ζ2519-ζ256-ζ2523-ζ252-ζ2522-ζ253-ζ2517-ζ258-ζ2513-ζ2512-ζ2518-ζ257-ζ2524-ζ25-ζ2521-ζ254    orthogonal faithful
ρ142200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2522+ζ253ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2518+ζ257ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2518+ζ257ζ2522+ζ253    orthogonal lifted from D25
ρ152200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2514+ζ2511ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2516+ζ259ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2516+ζ259ζ2514+ζ2511    orthogonal lifted from D25
ρ162200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2521+ζ254ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2524+ζ25ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2524+ζ25ζ2521+ζ254    orthogonal lifted from D25
ρ172-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2524+ζ25ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2519+ζ256-ζ2521-ζ254-ζ2516-ζ259-ζ2514-ζ2511-ζ2513-ζ2512-ζ2518-ζ257-ζ2523-ζ252-ζ2522-ζ253-ζ2517-ζ258-ζ2519-ζ256-ζ2524-ζ25    orthogonal faithful
ρ182200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2518+ζ257ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2517+ζ258ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2517+ζ258ζ2518+ζ257    orthogonal lifted from D25
ρ192-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2514+ζ2511ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2516+ζ259-ζ2519-ζ256-ζ2524-ζ25-ζ2521-ζ254-ζ2518-ζ257-ζ2523-ζ252-ζ2522-ζ253-ζ2517-ζ258-ζ2513-ζ2512-ζ2516-ζ259-ζ2514-ζ2511    orthogonal faithful
ρ202-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2518+ζ257ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2517+ζ258-ζ2522-ζ253-ζ2513-ζ2512-ζ2523-ζ252-ζ2516-ζ259-ζ2524-ζ25-ζ2514-ζ2511-ζ2521-ζ254-ζ2519-ζ256-ζ2517-ζ258-ζ2518-ζ257    orthogonal faithful
ρ212200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2517+ζ258ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2523+ζ252ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2523+ζ252ζ2517+ζ258    orthogonal lifted from D25
ρ222-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2522+ζ253ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2518+ζ257-ζ2513-ζ2512-ζ2523-ζ252-ζ2517-ζ258-ζ2514-ζ2511-ζ2521-ζ254-ζ2519-ζ256-ζ2516-ζ259-ζ2524-ζ25-ζ2518-ζ257-ζ2522-ζ253    orthogonal faithful
ρ232200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2519+ζ256ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2514+ζ2511ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2514+ζ2511ζ2519+ζ256    orthogonal lifted from D25
ρ242200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2523+ζ252ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2513+ζ2512ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2513+ζ2512ζ2523+ζ252    orthogonal lifted from D25
ρ252-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2513+ζ2512ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2522+ζ253-ζ2523-ζ252-ζ2517-ζ258-ζ2518-ζ257-ζ2519-ζ256-ζ2516-ζ259-ζ2524-ζ25-ζ2514-ζ2511-ζ2521-ζ254-ζ2522-ζ253-ζ2513-ζ2512    orthogonal faithful
ρ262200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2524+ζ25ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2519+ζ256ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2519+ζ256ζ2524+ζ25    orthogonal lifted from D25
ρ272200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2513+ζ2512ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2522+ζ253ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2522+ζ253ζ2513+ζ2512    orthogonal lifted from D25
ρ282-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2519+ζ256ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2514+ζ2511-ζ2524-ζ25-ζ2521-ζ254-ζ2516-ζ259-ζ2522-ζ253-ζ2517-ζ258-ζ2513-ζ2512-ζ2518-ζ257-ζ2523-ζ252-ζ2514-ζ2511-ζ2519-ζ256    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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