Show commands:
Magma
magma: G := TransitiveGroup(15, 78);
Group action invariants
| Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $78$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\wr S_5$ | ||
| CHM label: | $[3^{5}]S(5)=3wrS(5)$ | ||
| 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: | (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15), (1,4)(6,9)(11,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $120$: $S_5$ $360$: $S_5 \times C_3$ $9720$: 15T63 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $S_5$
Low degree siblings
15T78, 30T1016 x 2, 30T1033 x 2, 30T1038 x 2, 45T696 x 2, 45T723Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 108 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $29160=2^{3} \cdot 3^{6} \cdot 5$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 29160.h | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);