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G = D25order 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

C1C25 — D25
C1C5C25 — D25
C25 — D25
C1

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

25C2
5D5

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+5/2ζ2521254ζ252425ζ2519256ζ25142511ζ2518257ζ25132512ζ2517258ζ2522253ζ2523252ζ2516259    orthogonal faithful
ρ620-1+5/2-1-5/2ζ2517258ζ2523252ζ25132512ζ2522253ζ25142511ζ252425ζ2516259ζ2519256ζ2521254ζ2518257    orthogonal faithful
ρ720-1-5/2-1+5/2ζ252425ζ2519256ζ25142511ζ2516259ζ2517258ζ2522253ζ2523252ζ2518257ζ25132512ζ2521254    orthogonal faithful
ρ820-1+5/2-1-5/2ζ2522253ζ2518257ζ2517258ζ2523252ζ252425ζ2516259ζ2519256ζ2521254ζ25142511ζ25132512    orthogonal faithful
ρ920-1+5/2-1-5/2ζ2518257ζ2517258ζ2523252ζ25132512ζ2519256ζ2521254ζ25142511ζ252425ζ2516259ζ2522253    orthogonal faithful
ρ1020-1-5/2-1+5/2ζ25142511ζ2516259ζ2521254ζ252425ζ25132512ζ2517258ζ2522253ζ2523252ζ2518257ζ2519256    orthogonal faithful
ρ1120-1+5/2-1-5/2ζ2523252ζ25132512ζ2522253ζ2518257ζ2516259ζ2519256ζ2521254ζ25142511ζ252425ζ2517258    orthogonal faithful
ρ1220-1+5/2-1-5/2ζ25132512ζ2522253ζ2518257ζ2517258ζ2521254ζ25142511ζ252425ζ2516259ζ2519256ζ2523252    orthogonal faithful
ρ1320-1-5/2-1+5/2ζ2519256ζ25142511ζ2516259ζ2521254ζ2523252ζ2518257ζ25132512ζ2517258ζ2522253ζ252425    orthogonal faithful
ρ1420-1-5/2-1+5/2ζ2516259ζ2521254ζ252425ζ2519256ζ2522253ζ2523252ζ2518257ζ25132512ζ2517258ζ25142511    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);

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

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