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## G = C13⋊C3order 39 = 3·13

### The semidirect product of C13 and C3 acting faithfully

Aliases: C13⋊C3, SmallGroup(39,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C13⋊C3
 Chief series C1 — C13 — C13⋊C3
 Lower central C13 — C13⋊C3
 Upper central C1

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

Character table of C13⋊C3

 class 1 3A 3B 13A 13B 13C 13D size 1 13 13 3 3 3 3 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 1 1 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 1 1 linear of order 3 ρ4 3 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex faithful ρ5 3 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex faithful ρ6 3 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex faithful ρ7 3 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 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);`

C13⋊C3 is a maximal subgroup of   C13⋊C6  C13⋊A4  C91⋊C3  C914C3
C13⋊C3 is a maximal quotient of   C13⋊C9  C13⋊A4  C91⋊C3  C914C3

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

 0 2 0 2 1 2 2 0 1
,
 1 1 2 0 2 1 0 2 0
`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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