Show commands:
Magma
magma: G := TransitiveGroup(18, 284);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $284$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_6\wr S_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,13,10,2,14,9)(3,15,11,4,16,12)(5,18,7,6,17,8), (1,12,4,8,6,9,2,11,3,7,5,10)(13,15,17)(14,16,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $S_4$ $36$: $C_6\times S_3$ $48$: $S_4\times C_2$ $54$: $C_3^2 : C_6$ $72$: 12T45 $108$: 18T41 $144$: 18T61 $162$: $C_3 \wr S_3 $ $216$: 18T97 $324$: 18T119 $432$: 18T149 $648$: 18T203 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 \wr S_3 $
Low degree siblings
18T284 x 5, 36T1970 x 3, 36T1972 x 3, 36T1983 x 6, 36T1984 x 6, 36T2096 x 3Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 98 conjugacy class representatives for $C_6\wr S_3$
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: | 1296.1827 | magma: IdentifyGroup(G);
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| Character table: | 98 x 98 character table |
magma: CharacterTable(G);