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 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 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