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
| C5 — F5 |
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 |
►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 C13⋊3F5 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 C13⋊3F5 C13⋊F5 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 5T3 | x5-2 | 24·55 |
| 10T4 | x10+5x9-5x8-30x7+5x6+53x5+5x4-30x3-5x2+5x+1 | 28·38·511·74 |
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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