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## G = F7order 42 = 2·3·7

### Frobenius group

Aliases: F7, AGL1(𝔽7), C7⋊C6, D7⋊C3, C7⋊C3⋊C2, Aut(D7), Hol(C7), SmallGroup(42,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — F7
 Chief series C1 — C7 — C7⋊C3 — F7
 Lower central C7 — F7
 Upper central C1

Generators and relations for F7
G = < a,b | a7=b6=1, bab-1=a5 >

Character table of F7

 class 1 2 3A 3B 6A 6B 7 size 1 7 7 7 7 7 6 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ3 ζ32 1 linear of order 3 ρ4 1 1 ζ3 ζ32 ζ32 ζ3 1 linear of order 3 ρ5 1 -1 ζ3 ζ32 ζ6 ζ65 1 linear of order 6 ρ6 1 -1 ζ32 ζ3 ζ65 ζ6 1 linear of order 6 ρ7 6 0 0 0 0 0 -1 orthogonal faithful

Permutation representations of F7
On 7 points: primitive, sharply doubly transitive - transitive group 7T4
Generators in S7
```(1 2 3 4 5 6 7)
(2 4 3 7 5 6)```

`G:=sub<Sym(7)| (1,2,3,4,5,6,7), (2,4,3,7,5,6)>;`

`G:=Group( (1,2,3,4,5,6,7), (2,4,3,7,5,6) );`

`G=PermutationGroup([(1,2,3,4,5,6,7)], [(2,4,3,7,5,6)])`

`G:=TransitiveGroup(7,4);`

On 14 points - transitive group 14T4
Generators in S14
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 8)(2 11 3 14 5 13)(4 10 7 12 6 9)```

`G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9) );`

`G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,8),(2,11,3,14,5,13),(4,10,7,12,6,9)])`

`G:=TransitiveGroup(14,4);`

On 21 points - transitive group 21T4
Generators in S21
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 10 19)(2 13 21 7 14 17)(3 9 16 6 11 15)(4 12 18 5 8 20)```

`G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,10,19)(2,13,21,7,14,17)(3,9,16,6,11,15)(4,12,18,5,8,20)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,10,19)(2,13,21,7,14,17)(3,9,16,6,11,15)(4,12,18,5,8,20) );`

`G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,10,19),(2,13,21,7,14,17),(3,9,16,6,11,15),(4,12,18,5,8,20)])`

`G:=TransitiveGroup(21,4);`

F7 is a maximal subgroup of
C3⋊F7  D7⋊A4  C5⋊F7  C49⋊C6  C75F7  C73F7  C74F7  C7⋊F7  C72⋊C6  PGL2(𝔽7)  C11⋊F7
F7 is a maximal quotient of
C7⋊C12  C7⋊C18  C3⋊F7  D7⋊A4  C5⋊F7  C49⋊C6  C75F7  C73F7  C74F7  C7⋊F7  C72⋊C6  C11⋊F7

Polynomial with Galois group F7 over ℚ
actionf(x)Disc(f)
7T4x7-2-26·77
14T4x14-4x13+8x12-14x11+26x10-34x9+26x8-14x7+13x6+4x5-22x4+14x3+x2-4x+8-228·322·78·112·3832

Matrix representation of F7 in GL6(ℤ)

 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 1 0

`G:=sub<GL(6,Integers())| [0,0,0,0,0,-1,1,0,0,0,0,-1,0,1,0,0,0,-1,0,0,1,0,0,-1,0,0,0,1,0,-1,0,0,0,0,1,-1],[1,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,1,0,1,0,0,-1,0] >;`

F7 in GAP, Magma, Sage, TeX

`F_7`
`% in TeX`

`G:=Group("F7");`
`// GroupNames label`

`G:=SmallGroup(42,1);`
`// by ID`

`G=gap.SmallGroup(42,1);`
`# by ID`

`G:=PCGroup([3,-2,-3,-7,326,59]);`
`// Polycyclic`

`G:=Group<a,b|a^7=b^6=1,b*a*b^-1=a^5>;`
`// generators/relations`

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