# Properties

 Label 9T18 Degree $9$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2 : D_{6}$

# Related objects

## Group action invariants

 Degree $n$: $9$ Transitive number $t$: $18$ Group: $C_3^2 : D_{6}$ CHM label: $E(9):D_{12}=[3^{2}:2]S(3)=[1/2.S(3)^{2}]S(3)$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (1,2)(3,6)(4,8)(5,7), (3,4,5)(6,8,7), (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
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

## Low degree siblings

9T18, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 27T29, 36T87 x 2, 36T90

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

## Group invariants

 Order: $108=2^{2} \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [108, 17]
 Character table:  2 2 1 2 2 1 2 1 1 . 1 1 3 3 2 1 1 1 1 3 1 2 2 1 1a 3a 2a 2b 6a 2c 3b 6b 3c 3d 6c 2P 1a 3a 1a 1a 3a 1a 3b 3b 3c 3d 3d 3P 1a 1a 2a 2b 2c 2c 1a 2a 1a 1a 2b 5P 1a 3a 2a 2b 6a 2c 3b 6b 3c 3d 6c X.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 X.3 1 1 -1 1 -1 -1 1 -1 1 1 1 X.4 1 1 1 -1 -1 -1 1 1 1 1 -1 X.5 2 2 . -2 . . 2 . -1 -1 1 X.6 2 2 . 2 . . 2 . -1 -1 -1 X.7 2 -1 . . -1 2 2 . -1 2 . X.8 2 -1 . . 1 -2 2 . -1 2 . X.9 4 -2 . . . . 4 . 1 -2 . X.10 6 . -2 . . . -3 1 . . . X.11 6 . 2 . . . -3 -1 . . .