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## G = C13⋊C4order 52 = 22·13

### The semidirect product of C13 and C4 acting faithfully

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C13⋊C4
 Chief series C1 — C13 — D13 — C13⋊C4
 Lower central C13 — C13⋊C4
 Upper central C1

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

Character table of C13⋊C4

 class 1 2 4A 4B 13A 13B 13C size 1 13 13 13 4 4 4 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 1 linear of order 4 ρ4 1 -1 i -i 1 1 1 linear of order 4 ρ5 4 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal faithful ρ6 4 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal faithful ρ7 4 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 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 25)(2 20 13 17)(3 15 12 22)(4 23 11 14)(5 18 10 19)(6 26 9 24)(7 21 8 16)```

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

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

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

`G:=TransitiveGroup(26,4);`

C13⋊C4 is a maximal subgroup of   F13  C39⋊C4  C65⋊C4  C13⋊F5  C652C4  C91⋊C4  (C3×C39)⋊C4
C13⋊C4 is a maximal quotient of   C13⋊C8  C39⋊C4  C65⋊C4  C13⋊F5  C652C4  C91⋊C4  (C3×C39)⋊C4

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

 0 4 0 0 0 4 0 1 1 4 0 0 0 1 1 0
,
 1 2 4 0 0 1 4 1 0 3 3 0 0 0 3 0
`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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