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
| C13 — D26 |
Generators and relations for D26
G = < a,b | a26=b2=1, bab=a-1 >
Character table of D26
| class | 1 | 2A | 2B | 2C | 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 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 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 faithful |
| ρ6 | 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 faithful |
| 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 faithful |
| ρ8 | 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 |
| ρ9 | 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 faithful |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 faithful |
| 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 | orthogonal faithful |
| ρ14 | 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 |
| ρ15 | 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 |
| 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 | orthogonal lifted from D13 |
►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);
D26 is a maximal subgroup of
D52 C13⋊D4
D26 is a maximal quotient of Dic26 D52 C13⋊D4
Matrix representation of D26 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 47 | 24 |
| 0 | 20 | 17 |
| 1 | 0 | 0 |
| 0 | 43 | 51 |
| 0 | 23 | 10 |
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
Export
Subgroup lattice of D26 in TeX
Character table of D26 in TeX