Properties

Label 9T4
Order \(18\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_3$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $4$
Group :  $S_3\times C_3$
CHM label :  $S(3)[x]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9)(3,4,5)(6,7,8), (1,2)(4,5)(7,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Low degree siblings

6T5, 18T3

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

Group invariants

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

       1a 2a 3a  3b  6a 3c  3d  6b 3e
    2P 1a 1a 3a  3d  3e 3e  3b  3c 3c
    3P 1a 2a 1a  1a  2a 1a  1a  2a 1a
    5P 1a 2a 3a  3d  6b 3e  3b  6a 3c

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

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