Show commands:
Magma
magma: G := TransitiveGroup(42, 46);
Group action invariants
| Degree $n$: | $42$ | 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: | $C_7\times S_3^2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,23,2,24,3,22)(4,27,6,26,5,25)(7,30,9,28,8,29)(10,32,11,31,12,33)(13,36,15,35,14,34)(16,39,17,38,18,37)(19,41,21,40,20,42), (1,34,26,16,7,42,32,23,15,6,37,28,19,11,3,36,27,17,8,41,33,24,13,5,38,29,20,12,2,35,25,18,9,40,31,22,14,4,39,30,21,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $7$: $C_7$ $12$: $D_{6}$ x 2 $14$: $C_{14}$ x 3 $28$: 28T2 $36$: $S_3^2$ $42$: 21T6 x 2 $84$: 42T12 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3^2$
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 $C_7\times S_3^2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | 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: | 252.35 | magma: IdentifyGroup(G);
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| Character table: | 63 x 63 character table |
magma: CharacterTable(G);