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G = F5order 20 = 22·5

Frobenius group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C5 — F5
C1C5D5 — F5
C5 — F5
C1

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

5C2
5C4

Character table of F5

 class 124A4B5
 size 15554
ρ111111    trivial
ρ211-1-11    linear of order 2
ρ31-1-ii1    linear of order 4
ρ41-1i-i1    linear of order 4
ρ54000-1    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);

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

0100
0010
0001
-1-1-1-1
,
1000
0001
0100
-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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