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Properties

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

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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);
 
Nilpotency class:  $-1$ (not nilpotent)
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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 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);
 
Label:  14.1
magma: IdentifyGroup(G);
 
Character table:    
     2  1  1  .  .  .
     7  1  .  1  1  1

       1a 2a 7a 7b 7c
    2P 1a 1a 7b 7c 7a
    3P 1a 2a 7c 7a 7b
    5P 1a 2a 7b 7c 7a
    7P 1a 2a 1a 1a 1a

X.1     1  1  1  1  1
X.2     1 -1  1  1  1
X.3     2  .  A  B  C
X.4     2  .  B  C  A
X.5     2  .  C  A  B

A = E(7)+E(7)^6
B = E(7)^2+E(7)^5
C = E(7)^3+E(7)^4

magma: CharacterTable(G);