Show commands:
Magma
magma: G := TransitiveGroup(9, 33);
Group action invariants
| Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $A_9$ | ||
| CHM label: | $A9$ | ||
| 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));
| |
| Generators: | (1,2,3), (3,4,5,6,7,8,9) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Low degree siblings
36T23796Siblings 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, 1 $ | $1$ | $1$ | $()$ | |
| $ 7, 1, 1 $ | $25920$ | $7$ | $(1,8,5,4,7,9,3)$ | |
| $ 2, 2, 1, 1, 1, 1, 1 $ | $378$ | $2$ | $(5,6)(7,9)$ | |
| $ 3, 1, 1, 1, 1, 1, 1 $ | $168$ | $3$ | $(2,3,8)$ | |
| $ 4, 2, 1, 1, 1 $ | $7560$ | $4$ | $(1,4)(5,7,6,9)$ | |
| $ 3, 2, 2, 1, 1 $ | $7560$ | $6$ | $(2,8,3)(5,6)(7,9)$ | |
| $ 4, 3, 2 $ | $15120$ | $12$ | $(1,4)(2,3,8)(5,9,6,7)$ | |
| $ 3, 3, 3 $ | $2240$ | $3$ | $(1,2,4)(3,6,9)(5,8,7)$ | |
| $ 9 $ | $20160$ | $9$ | $(1,8,6,2,7,9,4,5,3)$ | |
| $ 5, 1, 1, 1, 1 $ | $3024$ | $5$ | $(3,4,5,6,9)$ | |
| $ 5, 3, 1 $ | $12096$ | $15$ | $(2,7,8)(3,5,9,4,6)$ | |
| $ 5, 3, 1 $ | $12096$ | $15$ | $(2,8,7)(3,5,9,4,6)$ | |
| $ 2, 2, 2, 2, 1 $ | $945$ | $2$ | $(1,9)(3,6)(4,5)(7,8)$ | |
| $ 3, 3, 1, 1, 1 $ | $3360$ | $3$ | $(1,4,7)(5,8,9)$ | |
| $ 6, 2, 1 $ | $30240$ | $6$ | $(1,8,4,9,7,5)(3,6)$ | |
| $ 9 $ | $20160$ | $9$ | $(1,4,6,3,2,7,9,5,8)$ | |
| $ 5, 2, 2 $ | $9072$ | $10$ | $(1,5,4,9,6)(2,3)(7,8)$ | |
| $ 4, 4, 1 $ | $11340$ | $4$ | $(1,4,2,3)(5,9,6,7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $181440=2^{6} \cdot 3^{4} \cdot 5 \cdot 7$ | 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: | 181440.b | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);