# Properties

 Label 9T19 Degree $9$ Order $144$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $(C_3^2:C_8):C_2$

# Related objects

## Group action invariants

 Degree $n$: $9$ Transitive number $t$: $19$ Group: $(C_3^2:C_8):C_2$ CHM label: $E(9):2D_{8}$ Parity: $-1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,6,4,5,2,3,8,7), (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $QD_{16}$

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1, 1$ $12$ $2$ $(3,7)(4,8)(5,6)$ $8, 1$ $18$ $8$ $(2,3,6,7,9,8,5,4)$ $4, 4, 1$ $36$ $4$ $(2,3,9,8)(4,5,7,6)$ $8, 1$ $18$ $8$ $(2,4,5,8,9,7,6,3)$ $4, 4, 1$ $18$ $4$ $(2,5,9,6)(3,4,8,7)$ $2, 2, 2, 2, 1$ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$ $6, 3$ $24$ $6$ $(1,2,3,4,8,6)(5,9,7)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

## Group invariants

 Order: $144=2^{4} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [144, 182]
 Character table:  2 4 2 3 2 3 3 4 1 1 3 2 1 . . . . . 1 2 1a 2a 8a 4a 8b 4b 2b 6a 3a 2P 1a 1a 4b 2b 4b 2b 1a 3a 3a 3P 1a 2a 8a 4a 8b 4b 2b 2a 1a 5P 1a 2a 8b 4a 8a 4b 2b 6a 3a 7P 1a 2a 8b 4a 8a 4b 2b 6a 3a X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 1 X.3 1 -1 1 -1 1 1 1 -1 1 X.4 1 1 -1 -1 -1 1 1 1 1 X.5 2 . . . . -2 2 . 2 X.6 2 . A . -A . -2 . 2 X.7 2 . -A . A . -2 . 2 X.8 8 -2 . . . . . 1 -1 X.9 8 2 . . . . . -1 -1 A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2