Show commands:
Magma
magma: G := TransitiveGroup(2, 1);
Group action invariants
| Degree $n$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2$ | ||
| CHM label: | $S2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
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| Nilpotency class: | $1$ | 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) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2 $ | $1$ | $2$ | $(1,2)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2$ (is prime) | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Label: | 2.1 | magma: IdentifyGroup(G);
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| Character table: |
2 1 1
1a 2a
2P 1a 1a
X.1 1 -1
X.2 1 1
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magma: CharacterTable(G);
Indecomposable integral representations
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Complete
list of indecomposable integral representations:
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