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Properties

Label 7T7
Degree $7$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_7$

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Show commands: Magma

magma: G := TransitiveGroup(7, 7);
 

Group action invariants

Degree $n$:  $7$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_7$
CHM label:   $S7$
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:  (1,2,3,4,5,6,7), (1,2)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T46, 21T38, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418

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 $2,1^{5}$ $21$ $2$ $1$ $(1,2)$
2B $2^{2},1^{3}$ $105$ $2$ $2$ $(1,2)(3,4)$
2C $2^{3},1$ $105$ $2$ $3$ $(1,2)(3,4)(5,6)$
3A $3,1^{4}$ $70$ $3$ $2$ $(1,2,3)$
3B $3^{2},1$ $280$ $3$ $4$ $(1,2,3)(4,5,6)$
4A $4,1^{3}$ $210$ $4$ $3$ $(1,2,3,4)$
4B $4,2,1$ $630$ $4$ $4$ $(1,2,3,4)(5,6)$
5A $5,1^{2}$ $504$ $5$ $4$ $(1,2,3,4,5)$
6A $3,2^{2}$ $210$ $6$ $4$ $(1,2,3)(4,5)(6,7)$
6B $3,2,1^{2}$ $420$ $6$ $3$ $(1,2,3)(4,5)$
6C $6,1$ $840$ $6$ $5$ $(1,2,3,4,5,6)$
7A $7$ $720$ $7$ $6$ $(1,2,3,4,5,6,7)$
10A $5,2$ $504$ $10$ $5$ $(1,2,3,4,5)(6,7)$
12A $4,3$ $420$ $12$ $5$ $(1,2,3,4)(5,6,7)$

Malle's constant $a(G)$:     $1$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5040.w
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 7A 10A 12A
Size 1 21 105 105 70 280 210 630 504 210 420 840 720 504 420
2 P 1A 1A 1A 1A 3A 3B 2B 2B 5A 3A 3A 3B 7A 5A 6A
3 P 1A 2A 2B 2C 1A 1A 4A 4B 5A 2B 2A 2C 7A 10A 4A
5 P 1A 2A 2B 2C 3A 3B 4A 4B 1A 6A 6B 6C 7A 2A 12A
7 P 1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 1A 10A 12A
Type

magma: CharacterTable(G);