Show commands:
Magma
magma: G := TransitiveGroup(21, 87);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $87$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:(C_6\times S_4)$ | ||
| 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: | (1,18,14,2,16,9)(3,21,11,7,20,12)(4,19,13,6,15,10)(5,17,8), (1,17,7,16,4,20,2,18,3,19,6,15)(5,21)(8,11,10)(9,13,14), (1,21,13,7,20,9,6,19,12,5,18,8,4,17,11,3,16,14,2,15,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $S_4$ $36$: $C_6\times S_3$ $48$: $S_4\times C_2$ $72$: 12T45 $144$: 18T61 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
28T544, 42T991, 42T992, 42T993, 42T994, 42T995, 42T996, 42T997, 42T998Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 51 conjugacy class representatives for $C_7^3:(C_6\times S_4)$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $49392=2^{4} \cdot 3^{2} \cdot 7^{3}$ | 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: | 49392.bg | magma: IdentifyGroup(G);
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| Character table: | 51 x 51 character table |
magma: CharacterTable(G);