Show commands:
Magma
magma: G := TransitiveGroup(16, 1672);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1672$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_4^4:\SL(2,3)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,3)(2,4)(5,16,12,6,15,11)(7,14,10)(8,13,9), (1,15,8,4,13,5,2,16,7,3,14,6)(9,11,10,12) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ $24$: $A_4\times C_2$, $\SL(2,3)$ x 2 $48$: 16T59 $96$: $C_2^4:C_6$ $192$: $C_2\wr A_4$ x 2, 24T291 $384$: 16T726 x 2, 16T729 $768$: 32T34813 $3072$: 48T? Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $A_4$
Degree 8: $C_2\wr A_4$
Low degree siblings
16T1672 x 7, 32T397446 x 4, 32T397447 x 4, 32T397448 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Conjugacy classes not computed
magma: ConjugacyClasses(G);
Group invariants
| Order: | $6144=2^{11} \cdot 3$ | 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: | 6144.da | magma: IdentifyGroup(G);
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| Character table: | 56 x 56 character table |
magma: CharacterTable(G);