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Magma
magma: G := TransitiveGroup(8, 14);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_4$ | ||
CHM label: | $S(4)[1/2]2=1/2(S_{4}[x]2)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,3)(2,8)(4,6)(5,7), (1,2,3)(5,6,7), (1,4)(2,6)(3,7)(5,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $S_4$
Low degree siblings
4T5, 6T7, 6T8, 12T8, 12T9, 24T10Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 1, 1 $ | $8$ | $3$ | $(2,3,8)(4,6,7)$ | |
$ 2, 2, 2, 2 $ | $3$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | |
$ 4, 4 $ | $6$ | $4$ | $(1,4,3,6)(2,5,8,7)$ | |
$ 2, 2, 2, 2 $ | $6$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $24=2^{3} \cdot 3$ | 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: | 24.12 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 3A | 4A | ||
Size | 1 | 3 | 6 | 8 | 6 | |
2 P | 1A | 1A | 1A | 3A | 2A | |
3 P | 1A | 2A | 2B | 1A | 4A | |
Type | ||||||
24.12.1a | R | |||||
24.12.1b | R | |||||
24.12.2a | R | |||||
24.12.3a | R | |||||
24.12.3b | R |
magma: CharacterTable(G);