metacyclic, supersoluble, monomial, Z-group
Aliases: F13, AGL1(𝔽13), C13⋊C12, D13.C6, C13⋊C3⋊C4, C13⋊C4⋊C3, C13⋊C6.C2, Aut(D13), Hol(C13), SmallGroup(156,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — F13 |
Generators and relations for F13
G = < a,b | a13=b12=1, bab-1=a6 >
Character table of F13
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 13 | |
| size | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | -i | i | -i | i | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | i | -i | i | -i | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | 1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | 1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | 1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | 1 | linear of order 12 |
| ρ13 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
►On 13 points: primitive, sharply doubly transitive - transitive group 13T6
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(2 12 5 6 4 8 13 3 10 9 11 7)
G:=sub<Sym(13)| (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12,5,6,4,8,13,3,10,9,11,7)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12,5,6,4,8,13,3,10,9,11,7) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13)], [(2,12,5,6,4,8,13,3,10,9,11,7)]])
G:=TransitiveGroup(13,6);
►On 26 points - transitive group 26T8
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 16)(2 14 5 21 4 23 13 18 10 24 11 22)(3 25 9 26 7 17 12 20 6 19 8 15)
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), (1,16)(2,14,5,21,4,23,13,18,10,24,11,22)(3,25,9,26,7,17,12,20,6,19,8,15)>;
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), (1,16)(2,14,5,21,4,23,13,18,10,24,11,22)(3,25,9,26,7,17,12,20,6,19,8,15) );
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)], [(1,16),(2,14,5,21,4,23,13,18,10,24,11,22),(3,25,9,26,7,17,12,20,6,19,8,15)]])
G:=TransitiveGroup(26,8);
F13 is a maximal subgroup of
C3⋊F13
F13 is a maximal quotient of C13⋊C24 C13⋊C36 C3⋊F13
| action | f(x) | Disc(f) |
|---|---|---|
| 13T6 | x13-65x11+1625x9-19500x7+113750x5-284375x3+203125x+69500 | 212·536·1313·2912 |
Matrix representation of F13 ►in GL12(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(12,Integers())| [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0] >;
F13 in GAP, Magma, Sage, TeX
F_{13} % in TeX
G:=Group("F13"); // GroupNames label
G:=SmallGroup(156,7);
// by ID
G=gap.SmallGroup(156,7);
# by ID
G:=PCGroup([4,-2,-3,-2,-13,24,1539,295,395]);
// Polycyclic
G:=Group<a,b|a^13=b^12=1,b*a*b^-1=a^6>;
// generators/relations
Export
Subgroup lattice of F13 in TeX
Character table of F13 in TeX