Copied to
clipboard

G = D13order 26 = 2·13

Dihedral group

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

Aliases: D13, C13⋊C2, sometimes denoted D26 or Dih13 or Dih26, SmallGroup(26,1)

Series: Derived Chief Lower central Upper central

C1C13 — D13
C1C13 — D13
C13 — D13
C1

Generators and relations for D13
 G = < a,b | a13=b2=1, bab=a-1 >

13C2

Character table of D13

 class 1213A13B13C13D13E13F
 size 113222222
ρ111111111    trivial
ρ21-1111111    linear of order 2
ρ320ζ139134ζ137136ζ138135ζ1310133ζ131213ζ1311132    orthogonal faithful
ρ420ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ137136    orthogonal faithful
ρ520ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ138135    orthogonal faithful
ρ620ζ137136ζ139134ζ131213ζ1311132ζ138135ζ1310133    orthogonal faithful
ρ720ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ131213    orthogonal faithful
ρ820ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ139134    orthogonal faithful

Permutation representations of D13
On 13 points: primitive - transitive group 13T2
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)

G:=sub<Sym(13)| (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8)])

G:=TransitiveGroup(13,2);

Regular action on 26 points - transitive group 26T2
Generators in S26
(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)
(1 16)(2 15)(3 14)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)

G:=sub<Sym(26)| (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), (1,16)(2,15)(3,14)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)>;

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), (1,16)(2,15)(3,14)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17) );

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)], [(1,16),(2,15),(3,14),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17)])

G:=TransitiveGroup(26,2);

Polynomial with Galois group D13 over ℚ
actionf(x)Disc(f)
13T2x13-x12-50x11-6x10+857x9+943x8-5045x7-9319x6+3890x5+13442x4+1835x3-2759x2+304x+4264·412·532·712·1492·81016

Matrix representation of D13 in GL2(𝔽53) generated by

2752
2852
,
4126
2112
G:=sub<GL(2,GF(53))| [27,28,52,52],[41,21,26,12] >;

D13 in GAP, Magma, Sage, TeX

D_{13}
% in TeX

G:=Group("D13");
// GroupNames label

G:=SmallGroup(26,1);
// by ID

G=gap.SmallGroup(26,1);
# by ID

G:=PCGroup([2,-2,-13,97]);
// Polycyclic

G:=Group<a,b|a^13=b^2=1,b*a*b=a^-1>;
// generators/relations

׿
×
𝔽