Show commands:
Magma
magma: G := TransitiveGroup(21, 46);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:S_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,8,20,7,11,18,6,14,16,5,10,21,4,13,19,3,9,17,2,12,15), (1,5)(2,4)(6,7)(8,16,9,19)(10,15,14,20)(11,18,13,17)(12,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
28T347, 42T538, 42T539, 42T540, 42T548Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 70 conjugacy class representatives for $C_7^3:S_4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $8232=2^{3} \cdot 3 \cdot 7^{3}$ | 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: | 8232.bt | magma: IdentifyGroup(G);
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| Character table: | 70 x 70 character table |
magma: CharacterTable(G);