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G = C13⋊C4order 52 = 22·13

The semidirect product of C13 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C13⋊C4, D13.C2, SmallGroup(52,3)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C4
C1C13D13 — C13⋊C4
C13 — C13⋊C4
C1

Generators and relations for C13⋊C4
 G = < a,b | a13=b4=1, bab-1=a5 >

13C2
13C4

Character table of C13⋊C4

 class 124A4B13A13B13C
 size 1131313444
ρ11111111    trivial
ρ211-1-1111    linear of order 2
ρ31-1-ii111    linear of order 4
ρ41-1i-i111    linear of order 4
ρ54000ζ139137136134ζ131213813513ζ13111310133132    orthogonal faithful
ρ64000ζ131213813513ζ13111310133132ζ139137136134    orthogonal faithful
ρ74000ζ13111310133132ζ139137136134ζ131213813513    orthogonal faithful

Permutation representations of C13⋊C4
On 13 points: primitive - transitive group 13T4
Generators in S13
(1 2 3 4 5 6 7 8 9 10 11 12 13)
(2 9 13 6)(3 4 12 11)(5 7 10 8)

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

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

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

G:=TransitiveGroup(13,4);

On 26 points - transitive group 26T4
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 17)(2 25 13 22)(3 20 12 14)(4 15 11 19)(5 23 10 24)(6 18 9 16)(7 26 8 21)

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

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

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

G:=TransitiveGroup(26,4);

Polynomial with Galois group C13⋊C4 over ℚ
actionf(x)Disc(f)
13T4x13-455x11+1300x10+76895x9-393900x8-5556850x7+39456950x6+152496175x5-1553609850x4-557344775x3+21836871550x2-10720383375x-99005292450248·318·524·1321·184612

Matrix representation of C13⋊C4 in GL4(𝔽5) generated by

0400
0401
1400
0110
,
1240
0141
0330
0030
G:=sub<GL(4,GF(5))| [0,0,1,0,4,4,4,1,0,0,0,1,0,1,0,0],[1,0,0,0,2,1,3,0,4,4,3,3,0,1,0,0] >;

C13⋊C4 in GAP, Magma, Sage, TeX

C_{13}\rtimes C_4
% in TeX

G:=Group("C13:C4");
// GroupNames label

G:=SmallGroup(52,3);
// by ID

G=gap.SmallGroup(52,3);
# by ID

G:=PCGroup([3,-2,-2,-13,6,290,221]);
// Polycyclic

G:=Group<a,b|a^13=b^4=1,b*a*b^-1=a^5>;
// generators/relations

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