Show commands:
Magma
magma: G := TransitiveGroup(8, 50);
Group action invariants
| Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_8$ | ||
| CHM label: | $S8$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
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| Nilpotency class: | $-1$ (not nilpotent) | magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,3,4,5,6,7,8), (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
Degree 2: None
Degree 4: None
Low degree siblings
16T1838, 28T502, 30T1153, 35T44Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 1, 1, 1, 1, 1, 1 $ | $28$ | $2$ | $(1,7)$ |
| $ 3, 3, 1, 1 $ | $1120$ | $3$ | $(2,8,6)(3,5,4)$ |
| $ 3, 3, 2 $ | $1120$ | $6$ | $(1,7)(2,6,8)(3,4,5)$ |
| $ 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $(3,5,4)$ |
| $ 3, 2, 1, 1, 1 $ | $1120$ | $6$ | $(1,7)(3,4,5)$ |
| $ 2, 2, 2, 2 $ | $105$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ |
| $ 6, 2 $ | $3360$ | $6$ | $(1,7)(2,4,8,3,6,5)$ |
| $ 2, 2, 1, 1, 1, 1 $ | $210$ | $2$ | $(1,7)(4,6)$ |
| $ 4, 2, 1, 1 $ | $2520$ | $4$ | $(1,4,7,6)(2,3)$ |
| $ 2, 2, 2, 1, 1 $ | $420$ | $2$ | $(1,7)(4,8)(5,6)$ |
| $ 6, 1, 1 $ | $3360$ | $6$ | $(2,4,6,5,8,3)$ |
| $ 3, 2, 2, 1 $ | $1680$ | $6$ | $(1,7)(2,6)(3,4,5)$ |
| $ 4, 4 $ | $1260$ | $4$ | $(1,5,2,6)(3,7,4,8)$ |
| $ 4, 2, 2 $ | $1260$ | $4$ | $(1,4,7,8)(2,3)(5,6)$ |
| $ 4, 1, 1, 1, 1 $ | $420$ | $4$ | $(2,5,6,3)$ |
| $ 4, 3, 1 $ | $3360$ | $12$ | $(1,8,4)(2,3,6,5)$ |
| $ 8 $ | $5040$ | $8$ | $(1,8,7,4,3,5,2,6)$ |
| $ 5, 1, 1, 1 $ | $1344$ | $5$ | $(2,5,6,8,7)$ |
| $ 5, 2, 1 $ | $4032$ | $10$ | $(2,8,5,7,6)(3,4)$ |
| $ 5, 3 $ | $2688$ | $15$ | $(1,8,2,7,6)(3,4,5)$ |
| $ 7, 1 $ | $5760$ | $7$ | $(2,7,4,6,3,5,8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ | 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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| Label: | 40320.a | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);