Show commands:
Magma
magma: G := TransitiveGroup(6, 15);
Group action invariants
Degree $n$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_6$ | ||
CHM label: | $A6$ | ||
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: | (1,2,3), (1,2)(3,4,5,6) | 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
6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 1, 1, 1 $ | $40$ | $3$ | $(4,5,6)$ |
$ 2, 2, 1, 1 $ | $45$ | $2$ | $(3,4)(5,6)$ |
$ 5, 1 $ | $72$ | $5$ | $(2,3,4,5,6)$ |
$ 5, 1 $ | $72$ | $5$ | $(2,3,4,6,5)$ |
$ 4, 2 $ | $90$ | $4$ | $(1,2)(3,4,5,6)$ |
$ 3, 3 $ | $40$ | $3$ | $(1,2,3)(4,5,6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $360=2^{3} \cdot 3^{2} \cdot 5$ | 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: | 360.118 | magma: IdentifyGroup(G);
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Character table: |
2 3 . 3 . . 2 . 3 2 2 . . . . 2 5 1 . . 1 1 . . 1a 3a 2a 5a 5b 4a 3b 2P 1a 3a 1a 5b 5a 2a 3b 3P 1a 1a 2a 5b 5a 4a 1a 5P 1a 3a 2a 1a 1a 4a 3b X.1 1 1 1 1 1 1 1 X.2 5 2 1 . . -1 -1 X.3 5 -1 1 . . -1 2 X.4 8 -1 . A *A . -1 X.5 8 -1 . *A A . -1 X.6 9 . 1 -1 -1 1 . X.7 10 1 -2 . . . 1 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);