metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic13, C13⋊2C4, C26.C2, C2.D13, SmallGroup(52,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — Dic13 |
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 |
►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 C13⋊3F5 C32⋊Dic13
Dic13p: Dic26 Dic39 Dic65 Dic91 ...
Dic13 is a maximal quotient of
C13⋊3F5 C32⋊Dic13
C2p.D13: C13⋊2C8 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
Export
Subgroup lattice of Dic13 in TeX
Character table of Dic13 in TeX