# Properties

 Label 9T3 Order $$18$$ n $$9$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{9}$

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## Group action invariants

 Degree $n$ : $9$ Transitive number $t$ : $3$ Group : $D_{9}$ CHM label : $D(9)=9:2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3,4,5,6,7,8,9), (1,8)(2,7)(3,6)(4,5) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

## Low degree siblings

18T5

Siblings 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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1$ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$ $9$ $2$ $9$ $(1,2,3,4,5,6,7,8,9)$ $9$ $2$ $9$ $(1,3,5,7,9,2,4,6,8)$ $3, 3, 3$ $2$ $3$ $(1,4,7)(2,5,8)(3,6,9)$ $9$ $2$ $9$ $(1,5,9,4,8,3,7,2,6)$

## Group invariants

 Order: $18=2 \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [18, 1]
 Character table: 2 1 1 . . . . 3 2 . 2 2 2 2 1a 2a 9a 9b 3a 9c 2P 1a 1a 9b 9c 3a 9a 3P 1a 2a 3a 3a 1a 3a 5P 1a 2a 9c 9a 3a 9b 7P 1a 2a 9b 9c 3a 9a X.1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 X.3 2 . -1 -1 2 -1 X.4 2 . A B -1 C X.5 2 . B C -1 A X.6 2 . C A -1 B A = E(9)^2+E(9)^7 B = E(9)^4+E(9)^5 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7