Properties

Label 9T2
Order \(9\)
n \(9\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_3^2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $2$
Group :  $C_3^2$
CHM label :  $E(9)=3[x]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

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

Group invariants

Order:  $9=3^{2}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [9, 2]
Character table:   
     3  2  2  2  2  2  2  2  2  2

       1a 3a 3b 3c 3d 3e 3f 3g 3h
    2P 1a 3h 3g 3f 3e 3d 3c 3b 3a
    3P 1a 1a 1a 1a 1a 1a 1a 1a 1a

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3