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## G = F5order 20 = 22·5

### Frobenius group

Aliases: F5, AGL1(𝔽5), C5⋊C4, D5.C2, Aut(D5), Hol(C5), Suzuki group Sz(2), SmallGroup(20,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5
 Chief series C1 — C5 — D5 — F5
 Lower central C5 — F5
 Upper central C1

Generators and relations for F5
G = < a,b | a5=b4=1, bab-1=a3 >

Character table of F5

 class 1 2 4A 4B 5 size 1 5 5 5 4 ρ1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 linear of order 2 ρ3 1 -1 -i i 1 linear of order 4 ρ4 1 -1 i -i 1 linear of order 4 ρ5 4 0 0 0 -1 orthogonal faithful

Permutation representations of F5
On 5 points: primitive, sharply doubly transitive - transitive group 5T3
Generators in S5
```(1 2 3 4 5)
(2 3 5 4)```

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

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

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

`G:=TransitiveGroup(5,3);`

On 10 points - transitive group 10T4
Generators in S10
```(1 2 3 4 5)(6 7 8 9 10)
(1 7)(2 9 5 10)(3 6 4 8)```

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

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

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

`G:=TransitiveGroup(10,4);`

Regular action on 20 points - transitive group 20T5
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 18 6 12)(2 20 10 15)(3 17 9 13)(4 19 8 11)(5 16 7 14)```

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

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

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

`G:=TransitiveGroup(20,5);`

F5 is a maximal subgroup of
C25⋊C4  S5  C32⋊F5  C24⋊F5
Cp⋊F5: C3⋊F5  D5.D5  C5⋊F5  C52⋊C4  C7⋊F5  C11⋊F5  C133F5  C13⋊F5 ...
F5 is a maximal quotient of
C5⋊C8  C25⋊C4  C32⋊F5  C24⋊F5
Cp⋊F5: C3⋊F5  D5.D5  C5⋊F5  C52⋊C4  C7⋊F5  C11⋊F5  C133F5  C13⋊F5 ...

Polynomial with Galois group F5 over ℚ
actionf(x)Disc(f)
5T3x5-224·55
10T4x10+5x9-5x8-30x7+5x6+53x5+5x4-30x3-5x2+5x+128·38·511·74

Matrix representation of F5 in GL4(ℤ) generated by

 0 1 0 0 0 0 1 0 0 0 0 1 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 1 0 1 0 0 -1 -1 -1 -1
`G:=sub<GL(4,Integers())| [0,0,0,-1,1,0,0,-1,0,1,0,-1,0,0,1,-1],[1,0,0,-1,0,0,1,-1,0,0,0,-1,0,1,0,-1] >;`

F5 in GAP, Magma, Sage, TeX

`F_5`
`% in TeX`

`G:=Group("F5");`
`// GroupNames label`

`G:=SmallGroup(20,3);`
`// by ID`

`G=gap.SmallGroup(20,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-5,6,74,77]);`
`// Polycyclic`

`G:=Group<a,b|a^5=b^4=1,b*a*b^-1=a^3>;`
`// generators/relations`

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