Properties

Label 4T4
Order \(12\)
n \(4\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $A_4$

Related objects

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Group action invariants

Degree $n$ :  $4$
Transitive number $t$ :  $4$
Group :  $A_4$
CHM label :  $A4$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3,4), (1,3,4)
$|\Aut(F/K)|$:  $1$

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

Group invariants

Order:  $12=2^{2} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [12, 3]
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