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Magma
magma: G := TransitiveGroup(18, 776);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $776$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_4^3.(C_2\times S_4)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7,13,6,12,17,2,8,14,5,11,18)(3,10,15,4,9,16), (1,5)(2,6)(7,16,8,15)(9,14,10,13)(11,17)(12,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ $48$: $S_4\times C_2$ $1296$: $S_3\wr S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: None
Degree 9: $S_3\wr S_3$
Low degree siblings
12T294, 18T770, 18T773, 18T777, 18T779, 18T780, 18T783, 18T785, 24T16593, 24T16594, 24T16595, 24T16596, 24T16597, 24T16598, 24T16599, 36T19182, 36T19183, 36T19184, 36T19185, 36T19192, 36T19193, 36T19194, 36T19195, 36T19211, 36T19212, 36T19213, 36T19214, 36T19215, 36T19216, 36T19219, 36T19220, 36T19221, 36T19222, 36T19229, 36T19230, 36T19231, 36T19232, 36T19247, 36T19248, 36T19249, 36T19250, 36T19251, 36T19252, 36T19254, 36T19255, 36T19256, 36T19257, 36T19420, 36T19421, 36T19422, 36T19423Siblings 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: | $82944=2^{10} \cdot 3^{4}$ | 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: | 82944.d | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);