Properties

Label 7T4
Degree $7$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_7$

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Show commands: Magma

magma: G := TransitiveGroup(7, 4);
 

Group action invariants

Degree $n$:  $7$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_7$
CHM label:   $F_{42}(7) = 7:6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,2,6,4,5), (1,2,3,4,5,6,7)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T4, 21T4, 42T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{7}$ $1$ $1$ $0$ $()$
2A $2^{3},1$ $7$ $2$ $3$ $(2,7)(3,6)(4,5)$
3A1 $3^{2},1$ $7$ $3$ $4$ $(2,5,3)(4,6,7)$
3A-1 $3^{2},1$ $7$ $3$ $4$ $(2,3,5)(4,7,6)$
6A1 $6,1$ $7$ $6$ $5$ $(2,6,5,7,3,4)$
6A-1 $6,1$ $7$ $6$ $5$ $(2,4,3,7,5,6)$
7A $7$ $6$ $7$ $6$ $(1,5,2,6,3,7,4)$

Malle's constant $a(G)$:     $1/3$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  42.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 7A
Size 1 7 7 7 7 7 6
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A
3 P 1A 2A 1A 1A 2A 2A 7A
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A
Type
42.1.1a R 1 1 1 1 1 1 1
42.1.1b R 1 1 1 1 1 1 1
42.1.1c1 C 1 1 ζ31 ζ3 ζ3 ζ31 1
42.1.1c2 C 1 1 ζ3 ζ31 ζ31 ζ3 1
42.1.1d1 C 1 1 ζ31 ζ3 ζ3 ζ31 1
42.1.1d2 C 1 1 ζ3 ζ31 ζ31 ζ3 1
42.1.6a R 6 0 0 0 0 0 1

magma: CharacterTable(G);