Group action invariants
Degree $n$: | $6$ | |
Transitive number $t$: | $6$ | |
Group: | $A_4\times C_2$ | |
CHM label: | $2A_{4}(6) = [2^{3}]3 = 2 wr 3$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $2$ | |
Generators: | (1,3,5)(2,4,6), (3,6) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Low degree siblings
8T13, 12T6, 12T7, 24T9Siblings 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, 1, 1, 1, 1 $ | $3$ | $2$ | $(3,6)$ |
$ 2, 2, 1, 1 $ | $3$ | $2$ | $(2,5)(3,6)$ |
$ 3, 3 $ | $4$ | $3$ | $(1,2,3)(4,5,6)$ |
$ 6 $ | $4$ | $6$ | $(1,2,3,4,5,6)$ |
$ 3, 3 $ | $4$ | $3$ | $(1,3,2)(4,6,5)$ |
$ 6 $ | $4$ | $6$ | $(1,3,5,4,6,2)$ |
$ 2, 2, 2 $ | $1$ | $2$ | $(1,4)(2,5)(3,6)$ |
Group invariants
Order: | $24=2^{3} \cdot 3$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
Label: | 24.13 |
Character table: |
2 3 3 3 1 1 1 1 3 3 1 . . 1 1 1 1 1 1a 2a 2b 3a 6a 3b 6b 2c 2P 1a 1a 1a 3b 3b 3a 3a 1a 3P 1a 2a 2b 1a 2c 1a 2c 2c 5P 1a 2a 2b 3b 6b 3a 6a 2c X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 -1 -1 X.3 1 -1 1 A -A /A -/A -1 X.4 1 -1 1 /A -/A A -A -1 X.5 1 1 1 A A /A /A 1 X.6 1 1 1 /A /A A A 1 X.7 3 1 -1 . . . . -3 X.8 3 -1 -1 . . . . 3 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |