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Properties

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

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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);
 
Nilpotency class:  $-1$ (not nilpotent)
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 $\leq 47$

Subfields

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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 1 $ $4$ $3$ $(2,3,4)$
$ 3, 1 $ $4$ $3$ $(2,4,3)$
$ 2, 2 $ $3$ $2$ $(1,2)(3,4)$

magma: ConjugacyClasses(G);
 

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);
 
Label:  12.3
magma: IdentifyGroup(G);
 
Character table:    
     2  2  .  .  2
     3  1  1  1  .

       1a 3a 3b 2a
    2P 1a 3b 3a 1a
    3P 1a 1a 1a 2a

X.1     1  1  1  1
X.2     1  A /A  1
X.3     1 /A  A  1
X.4     3  .  . -1

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3

magma: CharacterTable(G);