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G = C13⋊C6order 78 = 2·3·13

The semidirect product of C13 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C13⋊C6, D13⋊C3, C13⋊C3⋊C2, SmallGroup(78,1)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C6
C1C13C13⋊C3 — C13⋊C6
C13 — C13⋊C6
C1

Generators and relations for C13⋊C6
 G = < a,b | a13=b6=1, bab-1=a10 >

13C2
13C3
13C6

Character table of C13⋊C6

 class 123A3B6A6B13A13B
 size 1131313131366
ρ111111111    trivial
ρ21-111-1-111    linear of order 2
ρ31-1ζ32ζ3ζ65ζ611    linear of order 6
ρ411ζ3ζ32ζ32ζ311    linear of order 3
ρ511ζ32ζ3ζ3ζ3211    linear of order 3
ρ61-1ζ3ζ32ζ6ζ6511    linear of order 6
ρ7600000-1-13/2-1+13/2    orthogonal faithful
ρ8600000-1+13/2-1-13/2    orthogonal faithful

Permutation representations of C13⋊C6
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

Polynomial with Galois group C13⋊C6 over ℚ
actionf(x)Disc(f)
13T5x13-78x11+1989x9+1326x8-21255x7-33813x6+68328x5+216723x4+191178x3+51948x2-5850x-1875312·54·1322·472·8592·10972·87612·25804432

Matrix representation of C13⋊C6 in GL6(𝔽3)

011011
010101
020012
020010
020000
120021
,
111100
012002
022102
001100
002001
010211

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

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