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## G = C13⋊C6order 78 = 2·3·13

### The semidirect product of C13 and C6 acting faithfully

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C13⋊C6
 Chief series C1 — C13 — C13⋊C3 — C13⋊C6
 Lower central C13 — C13⋊C6
 Upper central C1

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

Character table of C13⋊C6

 class 1 2 3A 3B 6A 6B 13A 13B size 1 13 13 13 13 13 6 6 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 1 1 linear of order 2 ρ3 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 linear of order 6 ρ4 1 1 ζ3 ζ32 ζ32 ζ3 1 1 linear of order 3 ρ5 1 1 ζ32 ζ3 ζ3 ζ32 1 1 linear of order 3 ρ6 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 linear of order 6 ρ7 6 0 0 0 0 0 -1-√13/2 -1+√13/2 orthogonal faithful ρ8 6 0 0 0 0 0 -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)

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

`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`

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