Show commands:
Magma
magma: G := TransitiveGroup(3, 1);
Group action invariants
| Degree $n$: | $3$ | 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: | $C_3$ | ||
| CHM label: | $A3$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,3) | 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 $ | $1$ | $1$ | $()$ |
| $ 3 $ | $1$ | $3$ | $(1,2,3)$ |
| $ 3 $ | $1$ | $3$ | $(1,3,2)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $3$ (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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| Nilpotency class: | $1$ | ||
| Label: | 3.1 | magma: IdentifyGroup(G);
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| Character table: |
3 1 1 1
1a 3a 3b
X.1 1 1 1
X.2 1 A /A
X.3 1 /A A
A = E(3)
= (-1+Sqrt(-3))/2 = b3
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magma: CharacterTable(G);
Indecomposable integral representations
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Complete
list of indecomposable integral representations:
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