Show commands:
Magma
magma: G := TransitiveGroup(5, 1);
Group action invariants
Degree $n$: | $5$ | 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: | $C_5$ | ||
CHM label: | $C(5) = 5$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,4,5) | 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 $ | $1$ | $1$ | $()$ |
$ 5 $ | $1$ | $5$ | $(1,2,3,4,5)$ |
$ 5 $ | $1$ | $5$ | $(1,3,5,2,4)$ |
$ 5 $ | $1$ | $5$ | $(1,4,2,5,3)$ |
$ 5 $ | $1$ | $5$ | $(1,5,4,3,2)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $5$ (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: | 5.1 | magma: IdentifyGroup(G);
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Character table: |
5 1 1 1 1 1 1a 5a 5b 5c 5d X.1 1 1 1 1 1 X.2 1 A B /B /A X.3 1 B /A A /B X.4 1 /B A /A B X.5 1 /A /B B A A = E(5) B = E(5)^2 |
magma: CharacterTable(G);
Indecomposable integral representations
Complete
list of indecomposable integral representations:
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