Show commands:
Magma
magma: G := TransitiveGroup(21, 32);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7\wr S_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,18,12,3,16,10,5,21,8,7,19,13,2,17,11,4,15,9,6,20,14), (1,7,6,5,4,3,2)(8,19,13,17,11,15,9,20,14,18,12,16,10,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $7$: $C_7$ $14$: $C_{14}$ $42$: 21T6 $294$: 14T15 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
21T32 x 5, 42T269 x 6, 42T281 x 3, 42T282 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 140 conjugacy class representatives for $C_7\wr S_3$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2058=2 \cdot 3 \cdot 7^{3}$ | 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: | 2058.p | magma: IdentifyGroup(G);
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| Character table: | 140 x 140 character table |
magma: CharacterTable(G);