Show commands:
Magma
magma: G := TransitiveGroup(18, 485);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $485$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_6^3:S_4$ | ||
| 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: | (1,4)(2,3)(7,15,9,14,12,18,8,16,10,13,11,17), (1,15,12,5,14,8,4,18,10,2,16,11,6,13,7,3,17,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ x 3 $48$: $S_4\times C_2$ x 3 $96$: $V_4^2:S_3$ $192$: 12T100 $648$: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ $1296$: 18T301 $2592$: 18T402 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4\times C_2$
Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Low degree siblings
18T485 x 3, 36T5796 x 2, 36T5799 x 2, 36T5802, 36T5803 x 2, 36T5812 x 2, 36T5813 x 2, 36T5835 x 2, 36T5842, 36T5843 x 2, 36T5917 x 4, 36T5918 x 4, 36T5967 x 4, 36T5968 x 4, 36T6177 x 2, 36T6178 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 58 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5184=2^{6} \cdot 3^{4}$ | 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: | 5184.bu | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);