Properties

Label 9T17
Order \(81\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_3 \wr C_3 $

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $17$
Group :  $C_3 \wr C_3 $
CHM label :  $[3^{3}]3=3wr3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2,9), (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:  $C_3^2$
27:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T17 x 2, 27T19, 27T21, 27T27 x 3

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

Group invariants

Order:  $81=3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [81, 7]
Character table:   
      3  4   3   3   3  3  3   3  4   3   3  2  2  2  2  2  2  4

        1a  3a  3b  3c 3d 3e  3f 3g  3h  3i 3j 9a 9b 3k 9c 9d 3l
     2P 1a  3b  3a  3f 3e 3d  3c 3l  3i  3h 3k 9d 9c 3j 9b 9a 3g
     3P 1a  1a  1a  1a 1a 1a  1a 1a  1a  1a 1a 3g 3l 1a 3g 3l 1a
     5P 1a  3b  3a  3f 3e 3d  3c 3l  3i  3h 3k 9d 9c 3j 9b 9a 3g
     7P 1a  3a  3b  3c 3d 3e  3f 3g  3h  3i 3j 9a 9b 3k 9c 9d 3l

X.1      1   1   1   1  1  1   1  1   1   1  1  1  1  1  1  1  1
X.2      1   1   1   1  1  1   1  1   1   1  A  A  A /A /A /A  1
X.3      1   1   1   1  1  1   1  1   1   1 /A /A /A  A  A  A  1
X.4      1   A  /A  /A  1  1   A  1   A  /A  1  A /A  1  A /A  1
X.5      1  /A   A   A  1  1  /A  1  /A   A  1 /A  A  1 /A  A  1
X.6      1   A  /A  /A  1  1   A  1   A  /A  A /A  1 /A  1  A  1
X.7      1  /A   A   A  1  1  /A  1  /A   A /A  A  1  A  1 /A  1
X.8      1   A  /A  /A  1  1   A  1   A  /A /A  1  A  A /A  1  1
X.9      1  /A   A   A  1  1  /A  1  /A   A  A  1 /A /A  A  1  1
X.10     3   B  /B  -C  .  .   C /D -/B  -B  .  .  .  .  .  .  D
X.11     3  /B   B   C  .  .  -C  D  -B -/B  .  .  .  .  .  . /D
X.12     3   C  -C  -B  .  . -/B /D   B  /B  .  .  .  .  .  .  D
X.13     3  -C   C -/B  .  .  -B  D  /B   B  .  .  .  .  .  . /D
X.14     3  -B -/B   B  .  .  /B  D  -C   C  .  .  .  .  .  . /D
X.15     3 -/B  -B  /B  .  .   B /D   C  -C  .  .  .  .  .  .  D
X.16     3   .   .   .  D /D   .  3   .   .  .  .  .  .  .  .  3
X.17     3   .   .   . /D  D   .  3   .   .  .  .  .  .  .  .  3

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(3)-2*E(3)^2
  = (3+Sqrt(-3))/2 = 2+b3
C = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
D = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3