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
$\card{(G/N)}$ 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 | Index | Representative |
1A | $1^{4}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{2}$ | $1$ | $2$ | $2$ | $(1,3)(2,4)$ |
2B | $2^{2}$ | $2$ | $2$ | $2$ | $(1,2)(3,4)$ |
2C | $2,1^{2}$ | $2$ | $2$ | $1$ | $(2,4)$ |
4A | $4$ | $2$ | $4$ | $3$ | $(1,2,3,4)$ |
Malle's constant $a(G)$: $1$
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.