Show commands:
Magma
magma: G := TransitiveGroup(6, 12);
Group action invariants
Degree $n$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\PSL(2,5)$ | ||
CHM label: | $L(6) = PSL(2,5) = A_{5}(6)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4)(5,6), (1,2,3,4,6) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Low degree siblings
5T4, 10T7, 12T33, 15T5, 20T15, 30T9Siblings 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 | Representative |
$ 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 1, 1 $ | $15$ | $2$ | $(3,6)(4,5)$ | |
$ 5, 1 $ | $12$ | $5$ | $(2,3,5,4,6)$ | |
$ 5, 1 $ | $12$ | $5$ | $(2,4,3,6,5)$ | |
$ 3, 3 $ | $20$ | $3$ | $(1,2,3)(4,5,6)$ |
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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Nilpotency class: | not nilpotent | ||
Label: | 60.5 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 5A1 | 5A2 | ||
Size | 1 | 15 | 20 | 12 | 12 | |
2 P | 1A | 1A | 3A | 5A2 | 5A1 | |
3 P | 1A | 2A | 1A | 5A2 | 5A1 | |
5 P | 1A | 2A | 3A | 1A | 1A | |
Type | ||||||
60.5.1a | R | |||||
60.5.3a1 | R | |||||
60.5.3a2 | R | |||||
60.5.4a | R | |||||
60.5.5a | R |
magma: CharacterTable(G);