Show commands:
Magma
magma: G := TransitiveGroup(20, 673);
Group action invariants
| Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $673$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times C_4^4:S_5$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,4,2,3)(5,7,6,8)(9,13,17,10,14,18)(11,16,19,12,15,20), (1,9,3,11,2,10,4,12)(5,6)(7,8)(17,19,18,20), (1,7,9,3,5,11,2,8,10,4,6,12)(13,14)(15,16)(17,20,18,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $120$: $S_5$ $240$: $S_5\times C_2$ $1920$: $(C_2^4:A_5) : C_2$ $3840$: $C_2 \wr S_5$ $30720$: 20T568 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $S_5$
Degree 10: $(C_2^4:A_5) : C_2$
Low degree siblings
20T673, 40T19078 x 2, 40T19081 x 2, 40T19083 x 2, 40T19113 x 2, 40T19136, 40T19185 x 2, 40T19186 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 126 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $61440=2^{12} \cdot 3 \cdot 5$ | 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: | 61440.i | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);