Show commands:
Magma
magma: G := TransitiveGroup(21, 40);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:(C_3\times S_3)$ | ||
| 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: | (1,6,2)(4,5,7)(8,21,11,20,9,16)(10,18,12,15,13,17)(14,19), (1,19,8,2,15,11,3,18,14,4,21,10,5,17,13,6,20,9,7,16,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ $126$: 21T11 $882$: 14T26 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
21T40 x 5, 42T464 x 6, 42T473 x 3, 42T474 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 60 conjugacy class representatives for $C_7^3:(C_3\times S_3)$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $6174=2 \cdot 3^{2} \cdot 7^{3}$ | 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: | 6174.bn | magma: IdentifyGroup(G);
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| Character table: | 60 x 60 character table |
magma: CharacterTable(G);