# Properties

 Label 4T4 Degree $4$ Order $12$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $A_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(4, 4);

## Group action invariants

 Degree $n$: $4$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $A_4$ CHM label: $A4$ 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: (2,3,4), (1,3,4) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Degree 2: None

## Low degree siblings

6T4, 12T4

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 Index Representative 1A $1^{4}$ $1$ $1$ $0$ $()$ 2A $2^{2}$ $3$ $2$ $2$ $(1,2)(3,4)$ 3A1 $3,1$ $4$ $3$ $2$ $(1,2,3)$ 3A-1 $3,1$ $4$ $3$ $2$ $(1,3,2)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/2$

## Group invariants

 Order: $12=2^{2} \cdot 3$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 12.3 magma: IdentifyGroup(G); Character table:

 1A 2A 3A1 3A-1 Size 1 3 4 4 2 P 1A 1A 3A-1 3A1 3 P 1A 2A 1A 1A Type 12.3.1a R $1$ $1$ $1$ $1$ 12.3.1b1 C $1$ $1$ $ζ3−1$ $ζ3$ 12.3.1b2 C $1$ $1$ $ζ3$ $ζ3−1$ 12.3.3a R $3$ $−1$ $0$ $0$

magma: CharacterTable(G);