Show commands:
Magma
magma: G := TransitiveGroup(42, 20);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{21}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $21$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (4,5,6)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,35,36)(40,41,42), (1,5,7,12,13,18,21,22,26,29,33,34,37,40,3,6,8,11,14,17,19,24,27,28,31,35,38,41,2,4,9,10,15,16,20,23,25,30,32,36,39,42) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $7$: $C_7$ $14$: $C_{14}$ $18$: $S_3\times C_3$ $21$: $C_{21}$ $42$: 21T6, $C_{42}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 7: $C_7$
Degree 14: $C_{14}$
Degree 21: None
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 63 conjugacy class representatives for $S_3\times C_{21}$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $126=2 \cdot 3^{2} \cdot 7$ | 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: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 126.12 | magma: IdentifyGroup(G);
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| Character table: | 63 x 63 character table |
magma: CharacterTable(G);