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G = F7  order 42 = 2·3·7

Frobenius group

metacyclic, supersoluble, monomial, Z-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 >

C1
7C2
7C3
C7
7C6
D7
C7⋊C3
F7

Character table of F7

 class 123A3B6A6B7
 size 1777776
ρ11111111    trivial
ρ21-111-1-11    linear of order 2
ρ311ζ32ζ3ζ3ζ321    linear of order 3
ρ411ζ3ζ32ζ32ζ31    linear of order 3
ρ51-1ζ3ζ32ζ6ζ651    linear of order 6
ρ61-1ζ32ζ3ζ65ζ61    linear of order 6
ρ7600000-1    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  C7⋊5F7  C7⋊3F7  C7⋊4F7  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  C7⋊5F7  C7⋊3F7  C7⋊4F7  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)

010000
001000
000100
000010
000001
-1-1-1-1-1-1
,
100000
000001
000100
010000
-1-1-1-1-1-1
000010

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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Subgroup lattice of F7 in TeX
Character table of F7 in TeX

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