Properties

Label 9T1
Order \(9\)
n \(9\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_9$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $1$
Group :  $C_9$
CHM label :  $C(9)=9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,2,3,4,5,6,7,8,9)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 9 $ $1$ $9$ $(1,2,3,4,5,6,7,8,9)$
$ 9 $ $1$ $9$ $(1,3,5,7,9,2,4,6,8)$
$ 3, 3, 3 $ $1$ $3$ $(1,4,7)(2,5,8)(3,6,9)$
$ 9 $ $1$ $9$ $(1,5,9,4,8,3,7,2,6)$
$ 9 $ $1$ $9$ $(1,6,2,7,3,8,4,9,5)$
$ 3, 3, 3 $ $1$ $3$ $(1,7,4)(2,8,5)(3,9,6)$
$ 9 $ $1$ $9$ $(1,8,6,4,2,9,7,5,3)$
$ 9 $ $1$ $9$ $(1,9,8,7,6,5,4,3,2)$

Group invariants

Order:  $9=3^{2}$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [9, 1]
Character table:   
     3  2  2  2  2  2  2  2  2  2

       1a 9a 9b 3a 9c 9d 3b 9e 9f

X.1     1  1  1  1  1  1  1  1  1
X.2     1  A /A  1  A /A  1  A /A
X.3     1 /A  A  1 /A  A  1 /A  A
X.4     1  B /D  A  C /C /A  D /B
X.5     1  C /B  A  D /D /A  B /C
X.6     1  D /C  A  B /B /A  C /D
X.7     1 /D  C /A /B  B  A /C  D
X.8     1 /C  B /A /D  D  A /B  C
X.9     1 /B  D /A /C  C  A /D  B

A = E(3)
  = (-1+Sqrt(-3))/2 = b3
B = -E(9)^4-E(9)^7
C = E(9)^4
D = E(9)^7