# Properties

 Label 4T3 Degree $4$ Order $8$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_{4}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(4, 3);

## Group action invariants

 Degree $n$: $4$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $3$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_{4}$ CHM label: $D(4)$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $2$ magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,4), (1,3) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

## Low degree siblings

4T3, 8T4

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1$ $2$ $2$ $(2,4)$ $2, 2$ $2$ $2$ $(1,2)(3,4)$ $4$ $2$ $4$ $(1,2,3,4)$ $2, 2$ $1$ $2$ $(1,3)(2,4)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $8=2^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 8.3 magma: IdentifyGroup(G);
 Character table:  2 3 2 2 2 3 1a 2a 2b 4a 2c 2P 1a 1a 1a 2c 1a 3P 1a 2a 2b 4a 2c X.1 1 1 1 1 1 X.2 1 -1 -1 1 1 X.3 1 -1 1 -1 1 X.4 1 1 -1 -1 1 X.5 2 . . . -2 

magma: CharacterTable(G);