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Magma
magma: G := TransitiveGroup(34, 45);
Group invariants
| Abstract group: | $C_2^8.D_{34}$ | magma: IdentifyGroup(G);
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| Order: | $17408=2^{10} \cdot 17$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,21,8,28,13,34,19,5,25,12,32,18,3,24,10,29,16,2,22,7,27,14,33,20,6,26,11,31,17,4,23,9,30,15)$, $(1,31)(2,32)(3,29)(4,30)(5,27,6,28)(7,25)(8,26)(9,23)(10,24)(11,22)(12,21)(13,20,14,19)(15,17,16,18)(33,34)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $34$: $D_{17}$ $68$: $D_{34}$ $8704$: 34T30 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 17: $D_{17}$
Low degree siblings
34T45 x 29Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Conjugacy classes not computedmagma: ConjugacyClasses(G);
Character table
80 x 80 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed