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G = F13order 156 = 22·3·13

Frobenius group

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

C1C13 — F13
C1C13D13C13⋊C6 — F13
C13 — F13
C1

Generators and relations for F13
 G = < a,b | a13=b12=1, bab-1=a6 >

13C2
13C3
13C4
13C6
13C12

Character table of F13

 class 123A3B4A4B6A6B12A12B12C12D13
 size 1131313131313131313131312
ρ11111111111111    trivial
ρ21111-1-111-1-1-1-11    linear of order 2
ρ311ζ32ζ3-1-1ζ3ζ32ζ6ζ6ζ65ζ651    linear of order 6
ρ411ζ3ζ3211ζ32ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ511ζ3ζ32-1-1ζ32ζ3ζ65ζ65ζ6ζ61    linear of order 6
ρ611ζ32ζ311ζ3ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ71-111i-i-1-1-ii-ii1    linear of order 4
ρ81-111-ii-1-1i-ii-i1    linear of order 4
ρ91-1ζ32ζ3i-iζ65ζ6ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ31    linear of order 12
ρ101-1ζ3ζ32i-iζ6ζ65ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ321    linear of order 12
ρ111-1ζ3ζ32-iiζ6ζ65ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ321    linear of order 12
ρ121-1ζ32ζ3-iiζ65ζ6ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ31    linear of order 12
ρ131200000000000-1    orthogonal faithful

Permutation representations of F13
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 24)(2 22 5 16 4 18 13 26 10 19 11 17)(3 20 9 21 7 25 12 15 6 14 8 23)

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), (1,24)(2,22,5,16,4,18,13,26,10,19,11,17)(3,20,9,21,7,25,12,15,6,14,8,23)>;

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), (1,24)(2,22,5,16,4,18,13,26,10,19,11,17)(3,20,9,21,7,25,12,15,6,14,8,23) );

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)], [(1,24),(2,22,5,16,4,18,13,26,10,19,11,17),(3,20,9,21,7,25,12,15,6,14,8,23)])

G:=TransitiveGroup(26,8);

F13 is a maximal subgroup of   C3⋊F13
F13 is a maximal quotient of   C13⋊C24  C13⋊C36  C3⋊F13

Polynomial with Galois group F13 over ℚ
actionf(x)Disc(f)
13T6x13-65x11+1625x9-19500x7+113750x5-284375x3+203125x+69500212·536·1313·2912

Matrix representation of F13 in GL12(ℤ)

010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
-1-1-1-1-1-1-1-1-1-1-1-1
,
100000000000
000000100000
-1-1-1-1-1-1-1-1-1-1-1-1
000001000000
000000000001
000010000000
000000000010
000100000000
000000000100
001000000000
000000001000
010000000000

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

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