Group action invariants
| Degree $n$ : | $9$ | |
| Transitive number $t$ : | $2$ | |
| Group : | $C_3^2$ | |
| CHM label : | $E(9)=3[x]3$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $1$ | |
| Generators: | (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9) | |
| $|\Aut(F/K)|$: | $9$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Low degree siblings
There are no siblings 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, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,3,8)(2,4,6)(5,7,9)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,5,6)(2,3,7)(4,8,9)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,6,5)(2,7,3)(4,9,8)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,8,3)(2,6,4)(5,9,7)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,9,2)(3,5,4)(6,8,7)$ |
Group invariants
| Order: | $9=3^{2}$ | |
| Cyclic: | No | |
| Abelian: | Yes | |
| Solvable: | Yes | |
| GAP id: | [9, 2] |
| Character table: |
3 2 2 2 2 2 2 2 2 2
1a 3a 3b 3c 3d 3e 3f 3g 3h
2P 1a 3h 3g 3f 3e 3d 3c 3b 3a
3P 1a 1a 1a 1a 1a 1a 1a 1a 1a
X.1 1 1 1 1 1 1 1 1 1
X.2 1 1 A A A /A /A /A 1
X.3 1 1 /A /A /A A A A 1
X.4 1 A 1 A /A A /A 1 /A
X.5 1 /A 1 /A A /A A 1 A
X.6 1 A A /A 1 1 A /A /A
X.7 1 /A /A A 1 1 /A A A
X.8 1 A /A 1 A /A 1 A /A
X.9 1 /A A 1 /A A 1 /A A
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
|