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## G = Dic13order 52 = 22·13

### Dicyclic group

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — Dic13
 Chief series C1 — C13 — C26 — Dic13
 Lower central C13 — Dic13
 Upper central C1 — C2

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

Character table of Dic13

 class 1 2 4A 4B 13A 13B 13C 13D 13E 13F 26A 26B 26C 26D 26E 26F size 1 1 13 13 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 0 0 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 orthogonal lifted from D13 ρ6 2 2 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 orthogonal lifted from D13 ρ7 2 2 0 0 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 orthogonal lifted from D13 ρ8 2 2 0 0 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 orthogonal lifted from D13 ρ9 2 2 0 0 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 orthogonal lifted from D13 ρ10 2 2 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 orthogonal lifted from D13 ρ11 2 -2 0 0 ζ139+ζ134 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 -ζ1310-ζ133 -ζ139-ζ134 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 -ζ1311-ζ132 symplectic faithful, Schur index 2 ρ12 2 -2 0 0 ζ1312+ζ13 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 -ζ139-ζ134 -ζ1312-ζ13 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 -ζ137-ζ136 symplectic faithful, Schur index 2 ρ13 2 -2 0 0 ζ138+ζ135 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 -ζ137-ζ136 -ζ138-ζ135 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 -ζ139-ζ134 symplectic faithful, Schur index 2 ρ14 2 -2 0 0 ζ137+ζ136 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 -ζ1311-ζ132 -ζ137-ζ136 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 -ζ1310-ζ133 symplectic faithful, Schur index 2 ρ15 2 -2 0 0 ζ1311+ζ132 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 -ζ138-ζ135 -ζ1311-ζ132 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 -ζ1312-ζ13 symplectic faithful, Schur index 2 ρ16 2 -2 0 0 ζ1310+ζ133 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 -ζ1312-ζ13 -ζ1310-ζ133 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 -ζ138-ζ135 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 44 14 31)(2 43 15 30)(3 42 16 29)(4 41 17 28)(5 40 18 27)(6 39 19 52)(7 38 20 51)(8 37 21 50)(9 36 22 49)(10 35 23 48)(11 34 24 47)(12 33 25 46)(13 32 26 45)

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

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

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

Dic13 is a maximal subgroup of
C13⋊C8  C4×D13  C13⋊D4  C26.C6  C133F5  C32⋊Dic13
Dic13p: Dic26  Dic39  Dic65  Dic91 ...
Dic13 is a maximal quotient of
C133F5  C32⋊Dic13
C2p.D13: C132C8  Dic39  Dic65  Dic91 ...

Matrix representation of Dic13 in GL3(𝔽53) generated by

 52 0 0 0 26 52 0 1 0
,
 30 0 0 0 39 19 0 26 14
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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