Show commands:
Magma
magma: G := TransitiveGroup(21, 137);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $137$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^7.C_2\wr C_7:C_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,9,14,19,6,10,18)(2,8,15,21,4,12,17,3,7,13,20,5,11,16), (1,21,9,2,20,8)(3,19,7)(4,12,14,5,11,13)(6,10,15)(16,18,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ $168$: $C_2^3:(C_7: C_3)$ x 2 $336$: 14T18 x 2 $1344$: 14T35 $2688$: 14T44 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 7: $C_7:C_3$
Low degree siblings
42T2494, 42T2495 x 2, 42T2496 x 2, 42T2497, 42T2498, 42T2510Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 183 conjugacy class representatives for $C_3^7.C_2\wr C_7:C_3$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5878656=2^{7} \cdot 3^{8} \cdot 7$ | 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: | 5878656.e | magma: IdentifyGroup(G);
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| Character table: | 183 x 183 character table |
magma: CharacterTable(G);