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G = D25  order 50 = 2·52

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D25, C25⋊C2, C5.D5, sometimes denoted D50 or Dih25 or Dih50, SmallGroup(50,1)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for D25
 G = < a,b | a25=b2=1, bab=a-1 >

C1
25C2
C5
5D5
C25
D25

Character table of D25

 class 125A5B25A25B25C25D25E25F25G25H25I25J
 size 125222222222222
ρ111111111111111    trivial
ρ21-1111111111111    linear of order 2
ρ32022-1+√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
ρ42022-1-√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
ρ520-1-√5/2-1+√5/2ζ2521+ζ254ζ2524+ζ25ζ2519+ζ256ζ2514+ζ2511ζ2518+ζ257ζ2513+ζ2512ζ2517+ζ258ζ2522+ζ253ζ2523+ζ252ζ2516+ζ259    orthogonal faithful
ρ620-1+√5/2-1-√5/2ζ2517+ζ258ζ2523+ζ252ζ2513+ζ2512ζ2522+ζ253ζ2514+ζ2511ζ2524+ζ25ζ2516+ζ259ζ2519+ζ256ζ2521+ζ254ζ2518+ζ257    orthogonal faithful
ρ720-1-√5/2-1+√5/2ζ2524+ζ25ζ2519+ζ256ζ2514+ζ2511ζ2516+ζ259ζ2517+ζ258ζ2522+ζ253ζ2523+ζ252ζ2518+ζ257ζ2513+ζ2512ζ2521+ζ254    orthogonal faithful
ρ820-1+√5/2-1-√5/2ζ2522+ζ253ζ2518+ζ257ζ2517+ζ258ζ2523+ζ252ζ2524+ζ25ζ2516+ζ259ζ2519+ζ256ζ2521+ζ254ζ2514+ζ2511ζ2513+ζ2512    orthogonal faithful
ρ920-1+√5/2-1-√5/2ζ2518+ζ257ζ2517+ζ258ζ2523+ζ252ζ2513+ζ2512ζ2519+ζ256ζ2521+ζ254ζ2514+ζ2511ζ2524+ζ25ζ2516+ζ259ζ2522+ζ253    orthogonal faithful
ρ1020-1-√5/2-1+√5/2ζ2514+ζ2511ζ2516+ζ259ζ2521+ζ254ζ2524+ζ25ζ2513+ζ2512ζ2517+ζ258ζ2522+ζ253ζ2523+ζ252ζ2518+ζ257ζ2519+ζ256    orthogonal faithful
ρ1120-1+√5/2-1-√5/2ζ2523+ζ252ζ2513+ζ2512ζ2522+ζ253ζ2518+ζ257ζ2516+ζ259ζ2519+ζ256ζ2521+ζ254ζ2514+ζ2511ζ2524+ζ25ζ2517+ζ258    orthogonal faithful
ρ1220-1+√5/2-1-√5/2ζ2513+ζ2512ζ2522+ζ253ζ2518+ζ257ζ2517+ζ258ζ2521+ζ254ζ2514+ζ2511ζ2524+ζ25ζ2516+ζ259ζ2519+ζ256ζ2523+ζ252    orthogonal faithful
ρ1320-1-√5/2-1+√5/2ζ2519+ζ256ζ2514+ζ2511ζ2516+ζ259ζ2521+ζ254ζ2523+ζ252ζ2518+ζ257ζ2513+ζ2512ζ2517+ζ258ζ2522+ζ253ζ2524+ζ25    orthogonal faithful
ρ1420-1-√5/2-1+√5/2ζ2516+ζ259ζ2521+ζ254ζ2524+ζ25ζ2519+ζ256ζ2522+ζ253ζ2523+ζ252ζ2518+ζ257ζ2513+ζ2512ζ2517+ζ258ζ2514+ζ2511    orthogonal faithful

Permutation representations of D25
►On 25 points - transitive group 25T4
Generators in S25
(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)

(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)
 
G:=sub<Sym(25)| (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)>;
 
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14) );
 
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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14)]])
 
G:=TransitiveGroup(25,4);
 

D25 is a maximal subgroup of   C25⋊C4  D75  D125  C25⋊C10  C25⋊D5  D175
D25 is a maximal quotient of   Dic25  D75  D125  C25⋊D5  D175

Matrix representation of D25 ►in GL2(𝔽101) generated by

6897
1751
,
9964
112
G:=sub<GL(2,GF(101))| [68,17,97,51],[99,11,64,2] >;
 

D25 in GAP, Magma, Sage, TeX 

D_{25}
 
% in TeX
 
G:=Group("D25");
 
// GroupNames label
 
G:=SmallGroup(50,1);
 
// by ID
 
G=gap.SmallGroup(50,1);
 
# by ID
 
G:=PCGroup([3,-2,-5,-5,289,34,362]);
 
// Polycyclic
 
G:=Group<a,b|a^25=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D25 in TeX
Character table of D25 in TeX

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