# Properties

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

# Related objects

## 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
Generators:   (1,3,5)(2,4,6), (1,4)(2,5)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 3: 3T1

## Subfields

Degree 2: None

Degree 3: $C_3$

## Low degree siblings

4T4, 12T4
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1$ $3$ $2$ $(2,5)(3,6)$ $3, 3$ $4$ $3$ $(1,2,3)(4,5,6)$ $3, 3$ $4$ $3$ $(1,3,2)(4,6,5)$

## 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