# Properties

 Label 9T31 Degree $9$ Order $1296$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\wr S_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 31);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $31$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3\wr S_3$ CHM label: $[S(3)^{3}]S(3)=S(3)wrS(3)$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,9), (1,2), (3,6)(4,7)(5,8), (1,4,7)(2,5,8)(3,6,9) magma: Generators(G);

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $1296=2^{4} \cdot 3^{4}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 1296.3490 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);