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Properties

Label 9T5
Order \(18\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:C_2$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Low degree siblings

18T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 3, 3, 3 $ $2$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $2$ $3$ $(1,3,8)(2,4,6)(5,7,9)$
$ 3, 3, 3 $ $2$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 3, 3, 3 $ $2$ $3$ $(1,5,6)(2,3,7)(4,8,9)$

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [18, 4]
Character table:    
     2  1  1  .  .  .  .
     3  2  .  2  2  2  2

       1a 2a 3a 3b 3c 3d
    2P 1a 1a 3a 3b 3c 3d
    3P 1a 2a 1a 1a 1a 1a

X.1     1  1  1  1  1  1
X.2     1 -1  1  1  1  1
X.3     2  .  2 -1 -1 -1
X.4     2  . -1  2 -1 -1
X.5     2  . -1 -1 -1  2
X.6     2  . -1 -1  2 -1