Show commands:
Magma
magma: G := TransitiveGroup(4, 3);
Group action invariants
| Degree $n$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{4}$ | ||
| CHM label: | $D(4)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,3,4), (1,3) | 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: $C_2$
Low degree siblings
4T3, 8T4Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 1, 1 $ | $2$ | $2$ | $(2,4)$ | |
| $ 2, 2 $ | $2$ | $2$ | $(1,2)(3,4)$ | |
| $ 4 $ | $2$ | $4$ | $(1,2,3,4)$ | |
| $ 2, 2 $ | $1$ | $2$ | $(1,3)(2,4)$ |
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);
Additional information
If a degree four extension of fields has this, 4T3, as its Galois group, then there is a non-isomorphic degree four extension with the same normal closure. 4T3 give the lowest degree example of this phenomenon.