Show commands:
Magma
magma: G := TransitiveGroup(15, 32);
Group action invariants
Degree $n$: | $15$ | 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: | $C_5\wr S_3$ | ||
CHM label: | $[5^{3}]S(3)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,6,9,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $30$: $S_3 \times C_5$ $150$: $(C_5^2 : C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: None
Low degree siblings
15T32 x 3, 30T184 x 4, 30T185 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 65 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
Order: | $750=2 \cdot 3 \cdot 5^{3}$ | 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: | 750.26 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);