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Properties

Label 7T6
7T6 1 2 1->2 3 2->3 3->1 4 3->4 5 4->5 6 5->6 7 6->7 7->3
Degree $7$
Order $2520$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_7$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(7, 6);
 

Group invariants

Abstract group:  $A_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
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 $t$:  $6$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $A7$
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:  $(3,4,5,6,7)$, $(1,2,3)$
Copy content magma:Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

15T47 x 2, 21T33, 35T28, 42T294, 42T299

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 $2^{2},1^{3}$ $105$ $2$ $2$ $(1,2)(3,6)$
3A $3,1^{4}$ $70$ $3$ $2$ $(1,3,2)$
3B $3^{2},1$ $280$ $3$ $4$ $(2,6,7)(3,5,4)$
4A $4,2,1$ $630$ $4$ $4$ $(1,3,2,6)(4,7)$
5A $5,1^{2}$ $504$ $5$ $4$ $(3,6,4,7,5)$
6A $3,2^{2}$ $210$ $6$ $4$ $(1,2,3)(4,7)(5,6)$
7A1 $7$ $360$ $7$ $6$ $(1,3,7,5,2,6,4)$
7A-1 $7$ $360$ $7$ $6$ $(1,4,6,2,5,7,3)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 3B 4A 5A 6A 7A1 7A-1
Size 1 105 70 280 630 504 210 360 360
2 P 1A 1A 3A 3B 2A 5A 3A 7A1 7A-1
3 P 1A 2A 1A 1A 4A 5A 2A 7A-1 7A1
5 P 1A 2A 3A 3B 4A 1A 6A 7A-1 7A1
7 P 1A 2A 3A 3B 4A 5A 6A 1A 1A
Type
2520.a.1a R 1 1 1 1 1 1 1 1 1
2520.a.6a R 6 2 3 0 0 1 −1 −1 −1
2520.a.10a1 C 10 −2 1 1 0 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
2520.a.10a2 C 10 −2 1 1 0 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
2520.a.14a R 14 2 −1 2 0 −1 −1 0 0
2520.a.14b R 14 2 2 −1 0 −1 2 0 0
2520.a.15a R 15 −1 3 0 −1 0 −1 1 1
2520.a.21a R 21 1 −3 0 −1 1 1 0 0
2520.a.35a R 35 −1 −1 −1 1 0 −1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{7} - 2 x^{6} + \left(t + 9\right) x^{4} + t x^{3} + 2 x - 1$ Copy content Toggle raw display