Show commands:
Magma
magma: G := TransitiveGroup(8, 4);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_4$ | ||
CHM label: | $D_{8}(8)=[4]2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,8)(4,5,6,7), (1,6)(2,5)(3,4)(7,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Low degree siblings
4T3 x 2Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $8=2^{3}$ | 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: | $2$ | ||
Label: | 8.3 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A | ||
Size | 1 | 1 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 2A | |
Type | ||||||
8.3.1a | R | |||||
8.3.1b | R | |||||
8.3.1c | R | |||||
8.3.1d | R | |||||
8.3.2a | R |
magma: CharacterTable(G);