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Properties

Label 7T7
7T7 1 2 1->2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 7->1
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

Copy content magma:G := TransitiveGroup(7, 7);
 

Group invariants

Abstract group:  $S_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $S7$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3,4,5,6,7)$, $(1,2)$
Copy content 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 $2,1^{5}$ $21$ $2$ $1$ $(4,7)$
2B $2^{2},1^{3}$ $105$ $2$ $2$ $(2,6)(3,7)$
2C $2^{3},1$ $105$ $2$ $3$ $(1,6)(2,3)(4,5)$
3A $3,1^{4}$ $70$ $3$ $2$ $(1,4,5)$
3B $3^{2},1$ $280$ $3$ $4$ $(1,2,5)(3,4,6)$
4A $4,1^{3}$ $210$ $4$ $3$ $(2,7,6,3)$
4B $4,2,1$ $630$ $4$ $4$ $(1,2)(3,5,6,4)$
5A $5,1^{2}$ $504$ $5$ $4$ $(1,6,5,3,2)$
6A $3,2^{2}$ $210$ $6$ $4$ $(1,5,4)(2,6)(3,7)$
6B $3,2,1^{2}$ $420$ $6$ $3$ $(1,4)(2,7,3)$
6C $6,1$ $840$ $6$ $5$ $(1,4,2,6,5,3)$
7A $7$ $720$ $7$ $6$ $(1,6,4,2,7,5,3)$
10A $5,2$ $504$ $10$ $5$ $(1,3,6,2,5)(4,7)$
12A $4,3$ $420$ $12$ $5$ $(1,4,5)(2,3,6,7)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(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
5040.w.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5040.w.1b R 1 −1 1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1
5040.w.6a R 6 4 2 0 3 0 2 0 1 −1 1 0 −1 −1 −1
5040.w.6b R 6 −4 2 0 3 0 −2 0 1 −1 −1 0 −1 1 1
5040.w.14a R 14 6 2 2 2 −1 0 0 −1 2 0 −1 0 1 0
5040.w.14b R 14 −4 2 0 −1 2 2 0 −1 −1 −1 0 0 1 −1
5040.w.14c R 14 4 2 0 −1 2 −2 0 −1 −1 1 0 0 −1 1
5040.w.14d R 14 −6 2 −2 2 −1 0 0 −1 2 0 1 0 −1 0
5040.w.15a R 15 −5 −1 3 3 0 −1 −1 0 −1 1 0 1 0 −1
5040.w.15b R 15 5 −1 −3 3 0 1 −1 0 −1 −1 0 1 0 1
5040.w.20a R 20 0 −4 0 2 2 0 0 0 2 0 0 −1 0 0
5040.w.21a R 21 −1 1 3 −3 0 1 −1 1 1 −1 0 0 −1 1
5040.w.21b R 21 1 1 −3 −3 0 −1 −1 1 1 1 0 0 1 −1
5040.w.35a R 35 5 −1 1 −1 −1 −1 1 0 −1 −1 1 0 0 −1
5040.w.35b R 35 −5 −1 −1 −1 −1 1 1 0 −1 1 −1 0 0 1

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{7} + x + t$ Copy content Toggle raw display