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G = D26order 52 = 22·13

Dihedral group

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

Aliases: D26, C2×D13, C26⋊C2, C13⋊C22, sometimes denoted D52 or Dih26 or Dih52, SmallGroup(52,4)

Series: Derived Chief Lower central Upper central

C1C13 — D26
C1C13D13 — D26
C13 — D26
C1C2

Generators and relations for D26
 G = < a,b | a26=b2=1, bab=a-1 >

13C2
13C2
13C22

Character table of D26

 class 12A2B2C13A13B13C13D13E13F26A26B26C26D26E26F
 size 111313222222222222
ρ11111111111111111    trivial
ρ21-11-1111111-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111111111    linear of order 2
ρ41-1-11111111-1-1-1-1-1-1    linear of order 2
ρ52-200ζ139134ζ138135ζ137136ζ131213ζ1311132ζ131013313101331391341381351371361312131311132    orthogonal faithful
ρ62-200ζ137136ζ131213ζ139134ζ138135ζ1310133ζ131113213111321371361312131391341381351310133    orthogonal faithful
ρ72-200ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ13813513813513111321391341310133137136131213    orthogonal faithful
ρ82200ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ138135ζ138135ζ1311132ζ139134ζ1310133ζ137136ζ131213    orthogonal lifted from D13
ρ92-200ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ13913413913413121313111321381351310133137136    orthogonal faithful
ρ102200ζ137136ζ131213ζ139134ζ138135ζ1310133ζ1311132ζ1311132ζ137136ζ131213ζ139134ζ138135ζ1310133    orthogonal lifted from D13
ρ112200ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ139134ζ139134ζ131213ζ1311132ζ138135ζ1310133ζ137136    orthogonal lifted from D13
ρ122-200ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ13121313121313101331371361311132139134138135    orthogonal faithful
ρ132-200ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ13713613713613813513101331312131311132139134    orthogonal faithful
ρ142200ζ139134ζ138135ζ137136ζ131213ζ1311132ζ1310133ζ1310133ζ139134ζ138135ζ137136ζ131213ζ1311132    orthogonal lifted from D13
ρ152200ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ137136ζ137136ζ138135ζ1310133ζ131213ζ1311132ζ139134    orthogonal lifted from D13
ρ162200ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ131213ζ131213ζ1310133ζ137136ζ1311132ζ139134ζ138135    orthogonal lifted from D13

Permutation representations of D26
On 26 points - transitive group 26T3
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)

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

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

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

G:=TransitiveGroup(26,3);

Matrix representation of D26 in GL3(𝔽53) generated by

5200
04724
02017
,
100
04351
02310
G:=sub<GL(3,GF(53))| [52,0,0,0,47,20,0,24,17],[1,0,0,0,43,23,0,51,10] >;

D26 in GAP, Magma, Sage, TeX

D_{26}
% in TeX

G:=Group("D26");
// GroupNames label

G:=SmallGroup(52,4);
// by ID

G=gap.SmallGroup(52,4);
# by ID

G:=PCGroup([3,-2,-2,-13,434]);
// Polycyclic

G:=Group<a,b|a^26=b^2=1,b*a*b=a^-1>;
// generators/relations

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