Properties

 Label 9T2 Order $$9$$ n $$9$$ Cyclic No Abelian Yes Solvable Yes Primitive No $p$-group Yes Group: $C_3^2$

Related objects

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