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Magma
magma: G := TransitiveGroup(16, 1192);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1192$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4\wr C_4$ | ||
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: | (1,14,7,10,2,13,8,9)(3,15,6,12,4,16,5,11), (1,5,2,6)(3,7,4,8)(9,15,12,14,10,16,11,13) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 6, $C_2^2$ $8$: $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$ $16$: $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3 $32$: $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, 32T41 $64$: $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T74, 16T77, 16T121, 16T140, 16T154 $128$: 32T1095, 32T1311, 32T1313 $256$: 16T521, 16T563, 16T590 $512$: 32T13179 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $((C_8 : C_2):C_2):C_2$
Low degree siblings
16T1192 x 15, 32T36166 x 8, 32T36167 x 8, 32T36168 x 16, 32T36169 x 16, 32T36170 x 8, 32T56480 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 88 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
Order: | $1024=2^{10}$ | 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: | $6$ | ||
Label: | 1024.dgg | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);