# Properties

 Label 9T23 Order $$216$$ n $$9$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $(C_3^2:Q_8):C_3$

# Related objects

## Group action invariants

 Degree $n$ : $9$ Transitive number $t$ : $23$ Group : $(C_3^2:Q_8):C_3$ CHM label : $E(9):2A_{4}$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,9)(3,4,5)(6,7,8), (1,8,2,4)(3,5,6,7), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$
24:  $\SL(2,3)$

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

12T122, 24T562, 24T569, 27T82, 36T287, 36T309

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1$ $12$ $3$ $(3,4,5)(6,8,7)$ $3, 3, 1, 1, 1$ $12$ $3$ $(3,5,4)(6,7,8)$ $6, 2, 1$ $36$ $6$ $(2,3,5,9,8,6)(4,7)$ $4, 4, 1$ $54$ $4$ $(2,3,9,8)(4,5,7,6)$ $6, 2, 1$ $36$ $6$ $(2,6,8,9,5,3)(4,7)$ $2, 2, 2, 2, 1$ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$ $3, 3, 3$ $24$ $3$ $(1,2,3)(4,5,6)(7,8,9)$ $3, 3, 3$ $24$ $3$ $(1,2,6)(3,7,8)(4,5,9)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

## Group invariants

 Order: $216=2^{3} \cdot 3^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [216, 153]
 Character table:  2 3 1 1 1 2 1 3 . . . 3 3 2 2 1 . 1 1 2 2 3 1a 3a 3b 6a 4a 6b 2a 3c 3d 3e 2P 1a 3b 3a 3b 2a 3a 1a 3d 3c 3e 3P 1a 1a 1a 2a 4a 2a 2a 1a 1a 1a 5P 1a 3b 3a 6b 4a 6a 2a 3d 3c 3e X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A A 1 /A 1 /A A 1 X.3 1 /A A /A 1 A 1 A /A 1 X.4 2 -1 -1 1 . 1 -2 -1 -1 2 X.5 2 -A -/A A . /A -2 -/A -A 2 X.6 2 -/A -A /A . A -2 -A -/A 2 X.7 3 . . . -1 . 3 . . 3 X.8 8 2 2 . . . . -1 -1 -1 X.9 8 B /B . . . . -A -/A -1 X.10 8 /B B . . . . -/A -A -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3