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## G = F8order 56 = 23·7

### Frobenius group

Aliases: F8, AGL1(𝔽8), C23⋊C7, SmallGroup(56,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — F8
 Chief series C1 — C23 — F8
 Lower central C23 — F8
 Upper central C1

Generators and relations for F8
G = < a,b,c,d | a2=b2=c2=d7=1, ab=ba, ac=ca, dad-1=cb=bc, dbd-1=a, dcd-1=b >

Character table of F8

 class 1 2 7A 7B 7C 7D 7E 7F size 1 7 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 linear of order 7 ρ3 1 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 linear of order 7 ρ4 1 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 linear of order 7 ρ5 1 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 linear of order 7 ρ6 1 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 linear of order 7 ρ7 1 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 linear of order 7 ρ8 7 -1 0 0 0 0 0 0 orthogonal faithful

Permutation representations of F8
On 8 points: primitive, sharply doubly transitive - transitive group 8T25
Generators in S8
```(1 7)(2 3)(4 6)(5 8)
(1 8)(2 6)(3 4)(5 7)
(1 2)(3 7)(4 5)(6 8)
(2 3 4 5 6 7 8)```

`G:=sub<Sym(8)| (1,7)(2,3)(4,6)(5,8), (1,8)(2,6)(3,4)(5,7), (1,2)(3,7)(4,5)(6,8), (2,3,4,5,6,7,8)>;`

`G:=Group( (1,7)(2,3)(4,6)(5,8), (1,8)(2,6)(3,4)(5,7), (1,2)(3,7)(4,5)(6,8), (2,3,4,5,6,7,8) );`

`G=PermutationGroup([[(1,7),(2,3),(4,6),(5,8)], [(1,8),(2,6),(3,4),(5,7)], [(1,2),(3,7),(4,5),(6,8)], [(2,3,4,5,6,7,8)]])`

`G:=TransitiveGroup(8,25);`

On 14 points - transitive group 14T6
Generators in S14
```(2 13)(4 8)(5 9)(6 10)
(3 14)(5 9)(6 10)(7 11)
(1 12)(4 8)(6 10)(7 11)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)```

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

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

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

`G:=TransitiveGroup(14,6);`

On 28 points - transitive group 28T11
Generators in S28
```(2 25)(3 10)(4 27)(5 17)(6 18)(7 14)(9 21)(11 16)(12 28)(13 22)(15 26)(19 23)
(1 8)(3 26)(4 11)(5 28)(6 18)(7 19)(10 15)(12 17)(13 22)(14 23)(16 27)(20 24)
(1 20)(2 9)(4 27)(5 12)(6 22)(7 19)(8 24)(11 16)(13 18)(14 23)(17 28)(21 25)
(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 27 28)```

`G:=sub<Sym(28)| (2,25)(3,10)(4,27)(5,17)(6,18)(7,14)(9,21)(11,16)(12,28)(13,22)(15,26)(19,23), (1,8)(3,26)(4,11)(5,28)(6,18)(7,19)(10,15)(12,17)(13,22)(14,23)(16,27)(20,24), (1,20)(2,9)(4,27)(5,12)(6,22)(7,19)(8,24)(11,16)(13,18)(14,23)(17,28)(21,25), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)>;`

`G:=Group( (2,25)(3,10)(4,27)(5,17)(6,18)(7,14)(9,21)(11,16)(12,28)(13,22)(15,26)(19,23), (1,8)(3,26)(4,11)(5,28)(6,18)(7,19)(10,15)(12,17)(13,22)(14,23)(16,27)(20,24), (1,20)(2,9)(4,27)(5,12)(6,22)(7,19)(8,24)(11,16)(13,18)(14,23)(17,28)(21,25), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28) );`

`G=PermutationGroup([[(2,25),(3,10),(4,27),(5,17),(6,18),(7,14),(9,21),(11,16),(12,28),(13,22),(15,26),(19,23)], [(1,8),(3,26),(4,11),(5,28),(6,18),(7,19),(10,15),(12,17),(13,22),(14,23),(16,27),(20,24)], [(1,20),(2,9),(4,27),(5,12),(6,22),(7,19),(8,24),(11,16),(13,18),(14,23),(17,28),(21,25)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)]])`

`G:=TransitiveGroup(28,11);`

F8 is a maximal subgroup of   AΓL1(𝔽8)  C43⋊C7  C23.F8  C26⋊C7  C23⋊F8
F8 is a maximal quotient of   C7.F8  C43⋊C7  C23.F8  C26⋊C7  C23⋊F8

Polynomial with Galois group F8 over ℚ
actionf(x)Disc(f)
8T25x8-2x7-20x6+10x5+102x4+26x3-112x2-50x+7214·174·298
14T6x14+28x12+56x10-245x8-322x6+406x4-56x2-1214·724·318·674·8814

Matrix representation of F8 in GL7(ℤ)

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

`G:=sub<GL(7,Integers())| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;`

F8 in GAP, Magma, Sage, TeX

`F_8`
`% in TeX`

`G:=Group("F8");`
`// GroupNames label`

`G:=SmallGroup(56,11);`
`// by ID`

`G=gap.SmallGroup(56,11);`
`# by ID`

`G:=PCGroup([4,-7,-2,2,2,113,338,563]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^2=d^7=1,a*b=b*a,a*c=c*a,d*a*d^-1=c*b=b*c,d*b*d^-1=a,d*c*d^-1=b>;`
`// generators/relations`

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