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Magma
magma: G := TransitiveGroup(10, 7);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_{5}$ | ||
CHM label: | $A_{5}(10)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
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Nilpotency class: | $-1$ (not nilpotent) | magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9)(3,4)(5,10)(6,7), (1,3,5,7,9)(2,4,6,8,10) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: None
Low degree siblings
5T4, 6T12, 12T33, 15T5, 20T15, 30T9Siblings are shown 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, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 1 $ | $20$ | $3$ | $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$ |
$ 2, 2, 2, 2, 1, 1 $ | $15$ | $2$ | $( 2, 7)( 4,10)( 5, 8)( 6, 9)$ |
$ 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$ |
$ 5, 5 $ | $12$ | $5$ | $( 1, 2, 9, 6, 7)( 3, 8, 4,10, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $60=2^{2} \cdot 3 \cdot 5$ | 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: | no | magma: IsSolvable(G);
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Label: | 60.5 | magma: IdentifyGroup(G);
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Character table: |
2 2 . 2 . . 3 1 1 . . . 5 1 . . 1 1 1a 3a 2a 5a 5b 2P 1a 3a 1a 5b 5a 3P 1a 1a 2a 5b 5a 5P 1a 3a 2a 1a 1a X.1 1 1 1 1 1 X.2 3 . -1 A *A X.3 3 . -1 *A A X.4 4 1 . -1 -1 X.5 5 -1 1 . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);