Properties

Label 7T2
Degree $7$
Order $14$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_{7}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(7, 2);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{7}$ $1$ $1$ $()$
$2^{3},1$ $7$ $2$ $(2,7)(3,6)(4,5)$
$7$ $2$ $7$ $(1,2,3,4,5,6,7)$
$7$ $2$ $7$ $(1,3,5,7,2,4,6)$
$7$ $2$ $7$ $(1,4,7,3,6,2,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $14=2 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  14.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 7A1 7A2 7A3
Size 1 7 2 2 2
2 P 1A 1A 7A2 7A3 7A1
7 P 1A 2A 7A3 7A1 7A2
Type
14.1.1a R 1 1 1 1 1
14.1.1b R 1 1 1 1 1
14.1.2a1 R 2 0 ζ73+ζ73 ζ71+ζ7 ζ72+ζ72
14.1.2a2 R 2 0 ζ72+ζ72 ζ73+ζ73 ζ71+ζ7
14.1.2a3 R 2 0 ζ71+ζ7 ζ72+ζ72 ζ73+ζ73

magma: CharacterTable(G);