Show commands:
Magma
magma: G := TransitiveGroup(7, 7);
Group action invariants
| Degree $n$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_7$ | ||
| CHM label: | $S7$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,2,3,4,5,6,7), (1,2) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
14T46, 21T38, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $1^{7}$ | $1$ | $1$ | $()$ | |
| $2,1^{5}$ | $21$ | $2$ | $(3,4)$ | |
| $5,1^{2}$ | $504$ | $5$ | $(1,5,7,2,6)$ | |
| $5,2$ | $504$ | $10$ | $(1,2,5,6,7)(3,4)$ | |
| $2^{2},1^{3}$ | $105$ | $2$ | $(1,7)(3,4)$ | |
| $4,2,1$ | $630$ | $4$ | $(1,3,7,4)(2,6)$ | |
| $4,1^{3}$ | $210$ | $4$ | $(1,3,7,4)$ | |
| $2^{3},1$ | $105$ | $2$ | $(1,7)(2,5)(3,4)$ | |
| $3,1^{4}$ | $70$ | $3$ | $(2,6,5)$ | |
| $3,2^{2}$ | $210$ | $6$ | $(1,7)(2,5,6)(3,4)$ | |
| $4,3$ | $420$ | $12$ | $(1,3,7,4)(2,5,6)$ | |
| $3^{2},1$ | $280$ | $3$ | $(1,6,3)(2,5,4)$ | |
| $6,1$ | $840$ | $6$ | $(1,4,6,2,3,5)$ | |
| $7$ | $720$ | $7$ | $(1,7,3,2,4,5,6)$ | |
| $3,2,1^{2}$ | $420$ | $6$ | $(1,6,4)(5,7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 5040.w | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);