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G = C13⋊C3order 39 = 3·13

The semidirect product of C13 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C13⋊C3, SmallGroup(39,1)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C3
C1C13 — C13⋊C3
C13 — C13⋊C3
C1

Generators and relations for C13⋊C3
 G = < a,b | a13=b3=1, bab-1=a9 >

13C3

Character table of C13⋊C3

 class 13A3B13A13B13C13D
 size 113133333
ρ11111111    trivial
ρ21ζ32ζ31111    linear of order 3
ρ31ζ3ζ321111    linear of order 3
ρ4300ζ1311138137ζ13913313ζ136135132ζ13121310134    complex faithful
ρ5300ζ13121310134ζ1311138137ζ13913313ζ136135132    complex faithful
ρ6300ζ136135132ζ13121310134ζ1311138137ζ13913313    complex faithful
ρ7300ζ13913313ζ136135132ζ13121310134ζ1311138137    complex faithful

Permutation representations of C13⋊C3
On 13 points: primitive - transitive group 13T3
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)

G:=sub<Sym(13)| (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,4,10)(3,7,6)(5,13,11)(8,9,12)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,4,10)(3,7,6)(5,13,11)(8,9,12) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12)])

G:=TransitiveGroup(13,3);

Polynomial with Galois group C13⋊C3 over ℚ
actionf(x)Disc(f)
13T3x13-39x11+468x9-1989x7-507x6+2886x5+1443x4-624x3-234x2+3218·312·1316·594412·3584832

Matrix representation of C13⋊C3 in GL3(𝔽3) generated by

020
212
201
,
112
021
020
G:=sub<GL(3,GF(3))| [0,2,2,2,1,0,0,2,1],[1,0,0,1,2,2,2,1,0] >;

C13⋊C3 in GAP, Magma, Sage, TeX

C_{13}\rtimes C_3
% in TeX

G:=Group("C13:C3");
// GroupNames label

G:=SmallGroup(39,1);
// by ID

G=gap.SmallGroup(39,1);
# by ID

G:=PCGroup([2,-3,-13,37]);
// Polycyclic

G:=Group<a,b|a^13=b^3=1,b*a*b^-1=a^9>;
// generators/relations

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