Show commands:
Magma
magma: G := TransitiveGroup(9, 20);
Group action invariants
| Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3 \wr S_3 $ | ||
| CHM label: | $[3^{3}]S(3)=3wrS(3)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,9), (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) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Low degree siblings
9T20 x 2, 18T86 x 3, 27T37, 27T50 x 3, 27T70Siblings 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 $ | $3$ | $3$ | $(6,7,8)$ |
| $ 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 $ | $1$ | $3$ | $(1,9,2)(3,5,4)(6,8,7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $162=2 \cdot 3^{4}$ | 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: | 162.10 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);