Properties

Label 9T20
Order \(162\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3 \wr S_3 $

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $20$
Group :  $C_3 \wr S_3 $
CHM label :  $[3^{3}]S(3)=3wrS(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9), (3,6)(4,7)(5,8), (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:  $S_3$, $C_6$
18:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T20 x 2, 18T86 x 3, 27T37, 27T50 x 3, 27T70

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 $ $6$ $3$ $(3,4,5)(6,8,7)$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(3,5,4)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $ $9$ $2$ $(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $ $9$ $6$ $(3,6,4,7,5,8)$
$ 6, 1, 1, 1 $ $9$ $6$ $(3,6,5,8,4,7)$
$ 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, 2, 2, 2 $ $9$ $6$ $(1,2,9)(3,6)(4,7)(5,8)$
$ 6, 3 $ $9$ $6$ $(1,2,9)(3,6,4,7,5,8)$
$ 6, 3 $ $9$ $6$ $(1,2,9)(3,6,5,8,4,7)$
$ 3, 2, 2, 2 $ $9$ $6$ $(1,3)(2,4)(5,9)(6,8,7)$
$ 6, 3 $ $9$ $6$ $(1,3,2,4,9,5)(6,8,7)$
$ 3, 3, 3 $ $18$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $18$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 9 $ $18$ $9$ $(1,3,6,9,5,8,2,4,7)$
$ 6, 3 $ $9$ $6$ $(1,3,9,5,2,4)(6,8,7)$
$ 3, 3, 3 $ $1$ $3$ $(1,9,2)(3,5,4)(6,8,7)$

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 10]
Character table: Data not available.