metacyclic, supersoluble, monomial, Z-group
Aliases: C13⋊C6, D13⋊C3, C13⋊C3⋊C2, SmallGroup(78,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C6 |
Generators and relations for C13⋊C6
G = < a,b | a13=b6=1, bab-1=a10 >
Character table of C13⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 13A | 13B | |
| size | 1 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | linear of order 6 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | orthogonal faithful |
►On 13 points: primitive - transitive group 13T5
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(2 5 4 13 10 11)(3 9 7 12 6 8)
G:=sub<Sym(13)| (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,5,4,13,10,11)(3,9,7,12,6,8)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,5,4,13,10,11)(3,9,7,12,6,8) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13)], [(2,5,4,13,10,11),(3,9,7,12,6,8)]])
G:=TransitiveGroup(13,5);
►On 26 points - transitive group 26T6
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 14)(2 18 4 26 10 24)(3 22 7 25 6 21)(5 17 13 23 11 15)(8 16 9 20 12 19)
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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19)>;
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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19) );
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)], [(1,14),(2,18,4,26,10,24),(3,22,7,25,6,21),(5,17,13,23,11,15),(8,16,9,20,12,19)]])
G:=TransitiveGroup(26,6);
C13⋊C6 is a maximal subgroup of
F13 D39⋊C3 D13⋊A4 D65⋊C3
C13⋊C6 is a maximal quotient of C26.C6 C13⋊C18 D39⋊C3 D13⋊A4 D65⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 13T5 | x13-78x11+1989x9+1326x8-21255x7-33813x6+68328x5+216723x4+191178x3+51948x2-5850x-1875 | 312·54·1322·472·8592·10972·87612·25804432 |
Matrix representation of C13⋊C6 ►in GL6(𝔽3)
| 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 2 | 0 | 0 | 1 | 2 |
| 0 | 2 | 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 2 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 | 2 |
| 0 | 2 | 2 | 1 | 0 | 2 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 1 |
| 0 | 1 | 0 | 2 | 1 | 1 |
G:=sub<GL(6,GF(3))| [0,0,0,0,0,1,1,1,2,2,2,2,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,2,1,1,2,0,0,1],[1,0,0,0,0,0,1,1,2,0,0,1,1,2,2,1,2,0,1,0,1,1,0,2,0,0,0,0,0,1,0,2,2,0,1,1] >;
C13⋊C6 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_6 % in TeX
G:=Group("C13:C6"); // GroupNames label
G:=SmallGroup(78,1);
// by ID
G=gap.SmallGroup(78,1);
# by ID
G:=PCGroup([3,-2,-3,-13,650,86]);
// Polycyclic
G:=Group<a,b|a^13=b^6=1,b*a*b^-1=a^10>;
// generators/relations
Export
Subgroup lattice of C13⋊C6 in TeX
Character table of C13⋊C6 in TeX