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G = Dic13order 52 = 22·13

Dicyclic group

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

Aliases: Dic13, C132C4, C26.C2, C2.D13, SmallGroup(52,1)

Series: Derived Chief Lower central Upper central

C1C13 — Dic13
C1C13C26 — Dic13
C13 — Dic13
C1C2

Generators and relations for Dic13
 G = < a,b | a26=1, b2=a13, bab-1=a-1 >

13C4

Character table of Dic13

 class 124A4B13A13B13C13D13E13F26A26B26C26D26E26F
 size 111313222222222222
ρ11111111111111111    trivial
ρ211-1-1111111111111    linear of order 2
ρ31-1i-i111111-1-1-1-1-1-1    linear of order 4
ρ41-1-ii111111-1-1-1-1-1-1    linear of order 4
ρ52200ζ137136ζ131213ζ139134ζ138135ζ1310133ζ1311132ζ1311132ζ137136ζ131213ζ139134ζ138135ζ1310133    orthogonal lifted from D13
ρ62200ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ139134ζ139134ζ131213ζ1311132ζ138135ζ1310133ζ137136    orthogonal lifted from D13
ρ72200ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ138135ζ138135ζ1311132ζ139134ζ1310133ζ137136ζ131213    orthogonal lifted from D13
ρ82200ζ139134ζ138135ζ137136ζ131213ζ1311132ζ1310133ζ1310133ζ139134ζ138135ζ137136ζ131213ζ1311132    orthogonal lifted from D13
ρ92200ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ137136ζ137136ζ138135ζ1310133ζ131213ζ1311132ζ139134    orthogonal lifted from D13
ρ102200ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ131213ζ131213ζ1310133ζ137136ζ1311132ζ139134ζ138135    orthogonal lifted from D13
ρ112-200ζ139134ζ138135ζ137136ζ131213ζ1311132ζ131013313101331391341381351371361312131311132    symplectic faithful, Schur index 2
ρ122-200ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ13913413913413121313111321381351310133137136    symplectic faithful, Schur index 2
ρ132-200ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ13713613713613813513101331312131311132139134    symplectic faithful, Schur index 2
ρ142-200ζ137136ζ131213ζ139134ζ138135ζ1310133ζ131113213111321371361312131391341381351310133    symplectic faithful, Schur index 2
ρ152-200ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ13813513813513111321391341310133137136131213    symplectic faithful, Schur index 2
ρ162-200ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ13121313121313101331371361311132139134138135    symplectic faithful, Schur index 2

Smallest permutation representation of Dic13
Regular action on 52 points
Generators in S52
(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 51 52)
(1 52 14 39)(2 51 15 38)(3 50 16 37)(4 49 17 36)(5 48 18 35)(6 47 19 34)(7 46 20 33)(8 45 21 32)(9 44 22 31)(10 43 23 30)(11 42 24 29)(12 41 25 28)(13 40 26 27)

G:=sub<Sym(52)| (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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,52,14,39)(2,51,15,38)(3,50,16,37)(4,49,17,36)(5,48,18,35)(6,47,19,34)(7,46,20,33)(8,45,21,32)(9,44,22,31)(10,43,23,30)(11,42,24,29)(12,41,25,28)(13,40,26,27)>;

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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,52,14,39)(2,51,15,38)(3,50,16,37)(4,49,17,36)(5,48,18,35)(6,47,19,34)(7,46,20,33)(8,45,21,32)(9,44,22,31)(10,43,23,30)(11,42,24,29)(12,41,25,28)(13,40,26,27) );

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),(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,52,14,39),(2,51,15,38),(3,50,16,37),(4,49,17,36),(5,48,18,35),(6,47,19,34),(7,46,20,33),(8,45,21,32),(9,44,22,31),(10,43,23,30),(11,42,24,29),(12,41,25,28),(13,40,26,27)])

Matrix representation of Dic13 in GL3(𝔽53) generated by

5200
02652
010
,
3000
03919
02614
G:=sub<GL(3,GF(53))| [52,0,0,0,26,1,0,52,0],[30,0,0,0,39,26,0,19,14] >;

Dic13 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}
% in TeX

G:=Group("Dic13");
// GroupNames label

G:=SmallGroup(52,1);
// by ID

G=gap.SmallGroup(52,1);
# by ID

G:=PCGroup([3,-2,-2,-13,6,434]);
// Polycyclic

G:=Group<a,b|a^26=1,b^2=a^13,b*a*b^-1=a^-1>;
// generators/relations

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