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Properties

Label 6T4
Order \(12\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $4$
Group :  $A_4$
CHM label :  $A_{4}(6) = [2^{2}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,5), (1,3,5)(2,4,6)
$|\Aut(F/K)|$:  $2$

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: $C_3$

Low degree siblings

4T4, 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1 $ $3$ $2$ $(2,5)(3,6)$
$ 3, 3 $ $4$ $3$ $(1,2,3)(4,5,6)$
$ 3, 3 $ $4$ $3$ $(1,3,2)(4,6,5)$

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

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

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