# Properties

 Label 9T1 Order $$9$$ n $$9$$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group Yes Group: $C_9$

# Related objects

## 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