Show commands:
Magma
magma: G := TransitiveGroup(21, 139);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $139$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^7.S_7$ | ||
| 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: | (1,10,2,12,3,11)(4,17)(5,16)(6,18)(7,21,13)(8,20,14)(9,19,15), (1,4)(2,5)(3,6)(7,18,10,8,16,11,9,17,12)(19,20,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $5040$: $S_7$ $15120$: 21T56 $3674160$: 21T130 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 7: $S_7$
Low degree siblings
21T139, 42T2642 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 429 conjugacy class representatives for $C_3^7.S_7$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $11022480=2^{4} \cdot 3^{9} \cdot 5 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 11022480.a | magma: IdentifyGroup(G);
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| Character table: | 429 x 429 character table |
magma: CharacterTable(G);