metacyclic, supersoluble, monomial, Z-group
Aliases: C26.C6, C13⋊2C12, Dic13⋊C3, C13⋊C3⋊2C4, C2.(C13⋊C6), (C2×C13⋊C3).C2, SmallGroup(156,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C26.C6 |
Generators and relations for C26.C6
G = < a,b | a26=1, b6=a13, bab-1=a23 >
Character table of C26.C6
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 13A | 13B | 26A | 26B | |
| size | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 6 | 6 | |
| 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 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | i | -i | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | -i | i | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ13 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ14 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ15 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 1-√13/2 | 1+√13/2 | symplectic faithful, Schur index 2 |
| ρ16 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 1+√13/2 | 1-√13/2 | symplectic faithful, Schur index 2 |
►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 28 14 41)(2 45 17 40 10 51 15 32 4 27 23 38)(3 36 20 39 19 48 16 49 7 52 6 35)(5 44 26 37 11 42 18 31 13 50 24 29)(8 43 9 34 12 33 21 30 22 47 25 46)
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), (1,28,14,41)(2,45,17,40,10,51,15,32,4,27,23,38)(3,36,20,39,19,48,16,49,7,52,6,35)(5,44,26,37,11,42,18,31,13,50,24,29)(8,43,9,34,12,33,21,30,22,47,25,46)>;
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), (1,28,14,41)(2,45,17,40,10,51,15,32,4,27,23,38)(3,36,20,39,19,48,16,49,7,52,6,35)(5,44,26,37,11,42,18,31,13,50,24,29)(8,43,9,34,12,33,21,30,22,47,25,46) );
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)], [(1,28,14,41),(2,45,17,40,10,51,15,32,4,27,23,38),(3,36,20,39,19,48,16,49,7,52,6,35),(5,44,26,37,11,42,18,31,13,50,24,29),(8,43,9,34,12,33,21,30,22,47,25,46)]])
C26.C6 is a maximal subgroup of
C13⋊C24 Dic26⋊C3 C4×C13⋊C6 D26⋊C6 C39⋊3C12
C26.C6 is a maximal quotient of C13⋊2C24 C13⋊2C36 C39⋊3C12
Matrix representation of C26.C6 ►in GL6(𝔽3)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 2 | 1 | 2 | 0 |
| 0 | 0 | 0 | 2 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 2 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(3))| [0,1,0,0,0,1,1,1,0,0,0,2,0,0,2,2,0,0,0,0,2,1,2,0,0,0,0,2,1,0,0,1,0,0,0,2],[0,0,1,0,0,0,0,0,1,0,2,0,2,0,0,0,0,0,0,2,0,0,0,0,1,2,0,0,0,1,0,0,0,2,0,0] >;
C26.C6 in GAP, Magma, Sage, TeX
C_{26}.C_6 % in TeX
G:=Group("C26.C6"); // GroupNames label
G:=SmallGroup(156,1);
// by ID
G=gap.SmallGroup(156,1);
# by ID
G:=PCGroup([4,-2,-3,-2,-13,24,2307,295]);
// Polycyclic
G:=Group<a,b|a^26=1,b^6=a^13,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C26.C6 in TeX
Character table of C26.C6 in TeX