# Properties

 Label 9T22 Order $$162$$ n $$9$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $(C_3^2:C_3):C_2$

# Related objects

## Group action invariants

 Degree $n$ : $9$ Transitive number $t$ : $22$ Group : $(C_3^2:C_3):C_2$ CHM label : $[3^{3}:2]3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,9), (1,2)(4,5)(7,8), (1,4,7)(2,5,8)(3,6,9) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

## Low degree siblings

9T22 x 2, 18T85 x 3, 27T53 x 3, 27T62, 27T63

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1$ $6$ $3$ $(6,7,8)$ $3, 3, 1, 1, 1$ $6$ $3$ $(3,4,5)(6,7,8)$ $3, 3, 1, 1, 1$ $6$ $3$ $(3,4,5)(6,8,7)$ $2, 2, 2, 1, 1, 1$ $27$ $2$ $(2,9)(4,5)(7,8)$ $3, 3, 3$ $2$ $3$ $(1,2,9)(3,4,5)(6,7,8)$ $3, 3, 3$ $6$ $3$ $(1,2,9)(3,4,5)(6,8,7)$ $3, 3, 3$ $9$ $3$ $(1,3,6)(2,4,7)(5,8,9)$ $9$ $18$ $9$ $(1,3,6,2,4,7,9,5,8)$ $6, 3$ $27$ $6$ $(1,3,6)(2,5,7,9,4,8)$ $3, 3, 3$ $9$ $3$ $(1,6,3)(2,7,4)(5,9,8)$ $9$ $18$ $9$ $(1,6,4,2,7,5,9,8,3)$ $6, 3$ $27$ $6$ $(1,6,3)(2,8,4,9,7,5)$

## Group invariants

 Order: $162=2 \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [162, 11]
 Character table:  2 1 . . . 1 . . 1 . 1 1 . 1 3 4 3 3 3 1 4 3 2 2 1 2 2 1 1a 3a 3b 3c 2a 3d 3e 3f 9a 6a 3g 9b 6b 2P 1a 3a 3b 3c 1a 3d 3e 3g 9b 3g 3f 9a 3f 3P 1a 1a 1a 1a 2a 1a 1a 1a 3d 2a 1a 3d 2a 5P 1a 3a 3b 3c 2a 3d 3e 3g 9b 6b 3f 9a 6a 7P 1a 3a 3b 3c 2a 3d 3e 3f 9a 6a 3g 9b 6b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 X.3 1 1 1 1 -1 1 1 A A -A /A /A -/A X.4 1 1 1 1 -1 1 1 /A /A -/A A A -A X.5 1 1 1 1 1 1 1 A A A /A /A /A X.6 1 1 1 1 1 1 1 /A /A /A A A A X.7 2 -1 -1 2 . 2 -1 2 -1 . 2 -1 . X.8 2 -1 -1 2 . 2 -1 B -A . /B -/A . X.9 2 -1 -1 2 . 2 -1 /B -/A . B -A . X.10 6 . -3 . . -3 3 . . . . . . X.11 6 3 . . . -3 -3 . . . . . . X.12 6 -3 3 . . -3 . . . . . . . X.13 6 . . -3 . 6 . . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3