Show commands:
Magma
magma: G := TransitiveGroup(18, 556);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $556$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_6\wr S_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,10,2,9)(3,12,6,7)(4,11,5,8), (1,2)(3,6)(4,5)(13,14)(15,16)(17,18), (1,2)(3,4)(5,6), (13,15,18)(14,16,17), (1,14,10)(2,13,9)(3,18,8)(4,17,7)(5,15,12)(6,16,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 $24$: $S_4$ x 3, $S_3 \times C_2^2$ $48$: $S_4\times C_2$ x 9 $96$: $V_4^2:S_3$, 12T48 x 3 $192$: 12T100 x 3 $384$: 12T139 $1296$: $S_3\wr S_3$ $2592$: 18T394 $5184$: 18T483 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4\times C_2$
Degree 9: $S_3\wr S_3$
Low degree siblings
18T556 x 7, 36T8655 x 4, 36T8671 x 4, 36T8707 x 2, 36T8711 x 4, 36T8719 x 4, 36T8735 x 4, 36T8747 x 4, 36T8751 x 4, 36T8784 x 4, 36T8789 x 4, 36T8796 x 2, 36T8799 x 4, 36T8807 x 4, 36T8848 x 8, 36T8849 x 8, 36T8850 x 8, 36T8851 x 8, 36T8863 x 8, 36T8864 x 8, 36T8865 x 8, 36T8866 x 8, 36T8958 x 4, 36T8959 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 98 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $10368=2^{7} \cdot 3^{4}$ | 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: | 10368.bb | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);