Show commands:
Magma
magma: G := TransitiveGroup(21, 31);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:C_6$ | ||
| 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: | (1,16,11,4,19,9)(2,17,8,3,18,12)(5,20,13,7,15,14)(6,21,10), (1,12,19,6,11,21,4,10,16,2,9,18,7,8,20,5,14,15,3,13,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $14$: $D_{7}$ $42$: $F_7$ x 2, 21T3 $294$: 21T16 x 2, 21T19 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T31 x 11, 42T268 x 12Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 71 conjugacy class representatives for $C_7^3:C_6$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2058=2 \cdot 3 \cdot 7^{3}$ | 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: | 2058.bm | magma: IdentifyGroup(G);
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| Character table: | 71 x 71 character table |
magma: CharacterTable(G);