Properties

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

Related objects

Learn more about

Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $4$
Group :  $A_4$
CHM label :  $A_{4}(12)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,11,6)(2,9,7)(3,10,5)(4,8,12)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Low degree siblings

4T4, 6T4

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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$

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