Group action invariants
Degree $n$: | $6$ | |
Transitive number $t$: | $5$ | |
Group: | $S_3\times C_3$ | |
CHM label: | $F_{18}(6) = [3^{2}]2 = 3 wr 2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $3$ | |
Generators: | (1,4)(2,5)(3,6), (2,4,6) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Low degree siblings
9T4, 18T3Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 1, 1, 1 $ | $2$ | $3$ | $(2,4,6)$ |
$ 3, 1, 1, 1 $ | $2$ | $3$ | $(2,6,4)$ |
$ 2, 2, 2 $ | $3$ | $2$ | $(1,2)(3,4)(5,6)$ |
$ 6 $ | $3$ | $6$ | $(1,2,3,4,5,6)$ |
$ 6 $ | $3$ | $6$ | $(1,2,5,6,3,4)$ |
$ 3, 3 $ | $1$ | $3$ | $(1,3,5)(2,4,6)$ |
$ 3, 3 $ | $2$ | $3$ | $(1,3,5)(2,6,4)$ |
$ 3, 3 $ | $1$ | $3$ | $(1,5,3)(2,6,4)$ |
Group invariants
Order: | $18=2 \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
Label: | 18.3 |
Character table: |
2 1 . . 1 1 1 1 . 1 3 2 2 2 1 1 1 2 2 2 1a 3a 3b 2a 6a 6b 3c 3d 3e 2P 1a 3b 3a 1a 3c 3e 3e 3d 3c 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 5P 1a 3b 3a 2a 6b 6a 3e 3d 3c X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 X.3 1 A /A -1 -A -/A /A 1 A X.4 1 /A A -1 -/A -A A 1 /A X.5 1 A /A 1 A /A /A 1 A X.6 1 /A A 1 /A A A 1 /A X.7 2 -1 -1 . . . 2 -1 2 X.8 2 -/A -A . . . B -1 /B X.9 2 -A -/A . . . /B -1 B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 |