Properties

Label 9T31
Order \(1296\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\wr S_3$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $31$
Group :  $S_3\wr S_3$
CHM label :  $[S(3)^{3}]S(3)=S(3)wrS(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,9), (1,2), (3,6)(4,7)(5,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

12T213, 18T300, 18T303, 18T311, 18T312, 18T314, 18T315, 18T319, 18T320, 24T2893, 24T2894, 24T2895, 24T2912, 27T296, 27T298, 36T2197, 36T2198, 36T2199, 36T2201, 36T2202, 36T2210, 36T2211, 36T2212, 36T2213, 36T2214, 36T2215, 36T2216, 36T2217, 36T2218, 36T2219, 36T2220, 36T2225, 36T2226, 36T2229, 36T2305

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 $ $6$ $3$ $(1,2,9)$
$ 3, 3, 1, 1, 1 $ $12$ $3$ $(1,2,9)(3,4,5)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 2, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(2,9)$
$ 3, 2, 1, 1, 1, 1 $ $36$ $6$ $(2,9)(3,4,5)$
$ 3, 3, 2, 1 $ $36$ $6$ $(2,9)(3,4,5)(6,7,8)$
$ 2, 2, 1, 1, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)$
$ 3, 2, 2, 1, 1 $ $54$ $6$ $(2,9)(4,5)(6,7,8)$
$ 2, 2, 2, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 3, 3, 3 $ $72$ $3$ $(1,7,4)(2,8,5)(3,9,6)$
$ 9 $ $144$ $9$ $(1,7,4,2,8,5,9,6,3)$
$ 6, 3 $ $216$ $6$ $(1,7,4)(2,8,5,9,6,3)$
$ 2, 2, 2, 1, 1, 1 $ $18$ $2$ $(3,6)(4,7)(5,8)$
$ 3, 2, 2, 2 $ $36$ $6$ $(1,2,9)(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $ $36$ $6$ $(3,6,4,7,5,8)$
$ 6, 3 $ $72$ $6$ $(1,2,9)(3,6,4,7,5,8)$
$ 2, 2, 2, 2, 1 $ $54$ $2$ $(2,9)(3,6)(4,7)(5,8)$
$ 6, 2, 1 $ $108$ $6$ $(2,9)(3,6,4,7,5,8)$
$ 4, 2, 1, 1, 1 $ $54$ $4$ $(3,6)(4,7,5,8)$
$ 4, 3, 2 $ $108$ $12$ $(1,2,9)(3,6)(4,7,5,8)$
$ 4, 2, 2, 1 $ $162$ $4$ $(2,9)(3,6)(4,7,5,8)$

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1296, 3490]
Character table: Data not available.