Properties

Label 9T3
Order \(18\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{9}$

Related objects

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

18T5

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

Group invariants

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

       1a 2a 9a 9b 3a 9c
    2P 1a 1a 9b 9c 3a 9a
    3P 1a 2a 3a 3a 1a 3a
    5P 1a 2a 9c 9a 3a 9b
    7P 1a 2a 9b 9c 3a 9a

X.1     1  1  1  1  1  1
X.2     1 -1  1  1  1  1
X.3     2  . -1 -1  2 -1
X.4     2  .  A  B -1  C
X.5     2  .  B  C -1  A
X.6     2  .  C  A -1  B

A = E(9)^2+E(9)^7
B = E(9)^4+E(9)^5
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7