Show commands:
Magma
magma: G := TransitiveGroup(9, 31);
Group action invariants
| Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\wr S_3$ | ||
| CHM label: | $[S(3)^{3}]S(3)=S(3)wrS(3)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,9), (1,2), (3,6)(4,7)(5,8), (1,4,7)(2,5,8)(3,6,9) | magma: Generators(G);
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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, 36T2305Siblings 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);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 1296.3490 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);