Show commands:
Magma
magma: G := TransitiveGroup(27, 1204);
Group action invariants
| Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1204$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^7.C_3^3$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,5,26,3,4,27,2,6,25)(7,12,14,8,10,15,9,11,13)(16,19,22,17,20,23,18,21,24), (1,3,2)(4,5,6), (1,10,20,2,11,21,3,12,19)(4,14,23,6,13,22,5,15,24)(7,17,27)(8,18,26)(9,16,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 13 $9$: $C_3^2$ x 13 $27$: $C_9:C_3$ x 9, $C_3^2:C_3$ x 3, 27T4 $81$: $C_3 \wr C_3 $ x 9, 27T16 x 3, 27T18 $243$: 27T96 x 3, 27T99 x 3, 27T100 x 3, 27T105 x 3, 27T110 $2187$: 27T439 x 3, 27T441 x 3, 27T457 x 6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Degree 9: $C_3^2$
Low degree siblings
27T1204 x 242Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 825 conjugacy class representatives for $C_3^7.C_3^3$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $59049=3^{10}$ | 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: | $5$ | ||
| Label: | 59049.dj | magma: IdentifyGroup(G);
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| Character table: | not computed |
magma: CharacterTable(G);