Show commands:
Magma
magma: G := TransitiveGroup(21, 37);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:D_6$ | ||
| 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: | (1,3,5,7,2,4,6)(8,21,9,18,10,15,11,19,12,16,13,20,14,17), (1,10,15,6,8,16)(2,11,18,5,14,20)(3,12,21,4,13,17)(7,9,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $14$: $D_{7}$ $28$: $D_{14}$ $84$: 21T8 $588$: 14T25 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
21T37 x 5, 42T392 x 6, 42T393 x 6, 42T394 x 6, 42T400 x 3, 42T401 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 79 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $4116=2^{2} \cdot 3 \cdot 7^{3}$ | 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: | 4116.bu | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);