Properties

Label 9T7
Order \(27\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_3^2:C_3$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T7 x 3, 27T3

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

Group invariants

Order:  $27=3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [27, 3]
Character table:   
      3  3  2  2  3  2  2  2  2  2  2  3

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

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  A  A  A /A /A /A  1
X.3      1  1  1  1 /A /A /A  A  A  A  1
X.4      1  A /A  1  1  A /A  1  A /A  1
X.5      1 /A  A  1  1 /A  A  1 /A  A  1
X.6      1  A /A  1  A /A  1 /A  1  A  1
X.7      1 /A  A  1 /A  A  1  A  1 /A  1
X.8      1  A /A  1 /A  1  A  A /A  1  1
X.9      1 /A  A  1  A  1 /A /A  A  1  1
X.10     3  .  .  B  .  .  .  .  .  . /B
X.11     3  .  . /B  .  .  .  .  .  .  B

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