Group action invariants
| Degree $n$ : | $9$ | |
| Transitive number $t$ : | $1$ | |
| Group : | $C_9$ | |
| CHM label : | $C(9)=9$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $1$ | |
| Generators: | (1,2,3,4,5,6,7,8,9) | |
| $|\Aut(F/K)|$: | $9$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 9 $ | $1$ | $9$ | $(1,2,3,4,5,6,7,8,9)$ |
| $ 9 $ | $1$ | $9$ | $(1,3,5,7,9,2,4,6,8)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ |
| $ 9 $ | $1$ | $9$ | $(1,5,9,4,8,3,7,2,6)$ |
| $ 9 $ | $1$ | $9$ | $(1,6,2,7,3,8,4,9,5)$ |
| $ 3, 3, 3 $ | $1$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ |
| $ 9 $ | $1$ | $9$ | $(1,8,6,4,2,9,7,5,3)$ |
| $ 9 $ | $1$ | $9$ | $(1,9,8,7,6,5,4,3,2)$ |
Group invariants
| Order: | $9=3^{2}$ | |
| Cyclic: | Yes | |
| Abelian: | Yes | |
| Solvable: | Yes | |
| GAP id: | [9, 1] |
| Character table: |
3 2 2 2 2 2 2 2 2 2
1a 9a 9b 3a 9c 9d 3b 9e 9f
X.1 1 1 1 1 1 1 1 1 1
X.2 1 A /A 1 A /A 1 A /A
X.3 1 /A A 1 /A A 1 /A A
X.4 1 B /D A C /C /A D /B
X.5 1 C /B A D /D /A B /C
X.6 1 D /C A B /B /A C /D
X.7 1 /D C /A /B B A /C D
X.8 1 /C B /A /D D A /B C
X.9 1 /B D /A /C C A /D B
A = E(3)
= (-1+Sqrt(-3))/2 = b3
B = -E(9)^4-E(9)^7
C = E(9)^4
D = E(9)^7
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