# Properties

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

# Related objects

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

 Label Cycle Type Size Order Representative $1^{7}$ $1$ $1$ $()$ $2^{3},1$ $7$ $2$ $(2,7)(3,6)(4,5)$ $7$ $2$ $7$ $(1,2,3,4,5,6,7)$ $7$ $2$ $7$ $(1,3,5,7,2,4,6)$ $7$ $2$ $7$ $(1,4,7,3,6,2,5)$

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 $2$ $0$ $ζ7−3+ζ73$ $ζ7−1+ζ7$ $ζ7−2+ζ72$ 14.1.2a2 R $2$ $0$ $ζ7−2+ζ72$ $ζ7−3+ζ73$ $ζ7−1+ζ7$ 14.1.2a3 R $2$ $0$ $ζ7−1+ζ7$ $ζ7−2+ζ72$ $ζ7−3+ζ73$

magma: CharacterTable(G);