Show commands:
Magma
magma: G := TransitiveGroup(9, 34);
Group action invariants
| Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $S_9$ | ||
| CHM label: | $S9$ | ||
| 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), (1,2,3,4,5,6,7,8,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Low degree siblings
18T887, 36T28590Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 1 $ | $945$ | $2$ | $(1,4)(2,6)(3,9)(5,7)$ | |
| $ 4, 4, 1 $ | $11340$ | $4$ | $(1,5,4,7)(2,9,6,3)$ | |
| $ 8, 1 $ | $45360$ | $8$ | $(1,2,5,9,4,6,7,3)$ | |
| $ 2, 2, 1, 1, 1, 1, 1 $ | $378$ | $2$ | $(1,4)(5,7)$ | |
| $ 4, 2, 2, 1 $ | $11340$ | $4$ | $(1,5,4,7)(2,6)(3,9)$ | |
| $ 2, 2, 2, 1, 1, 1 $ | $1260$ | $2$ | $(1,4)(2,6)(5,7)$ | |
| $ 2, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $(3,8)$ | |
| $ 3, 3, 1, 1, 1 $ | $3360$ | $3$ | $(1,5,2)(4,7,6)$ | |
| $ 3, 3, 2, 1 $ | $10080$ | $6$ | $(1,2,5)(3,8)(4,6,7)$ | |
| $ 6, 1, 1, 1 $ | $10080$ | $6$ | $(1,6,5,4,2,7)$ | |
| $ 3, 3, 3 $ | $2240$ | $3$ | $(1,2,5)(3,9,8)(4,6,7)$ | |
| $ 6, 3 $ | $20160$ | $6$ | $(1,7,2,4,5,6)(3,8,9)$ | |
| $ 3, 1, 1, 1, 1, 1, 1 $ | $168$ | $3$ | $(3,8,9)$ | |
| $ 3, 2, 2, 2 $ | $2520$ | $6$ | $(1,4)(2,6)(3,9,8)(5,7)$ | |
| $ 6, 2, 1 $ | $30240$ | $6$ | $(1,8,9,7,2,6)(4,5)$ | |
| $ 5, 1, 1, 1, 1 $ | $3024$ | $5$ | $(1,7,6,2,5)$ | |
| $ 5, 3, 1 $ | $24192$ | $15$ | $(1,6,5,7,2)(3,9,8)$ | |
| $ 5, 2, 1, 1 $ | $18144$ | $10$ | $(1,2,7,5,6)(3,9)$ | |
| $ 5, 2, 2 $ | $9072$ | $10$ | $(1,6,5,7,2)(3,9)(4,8)$ | |
| $ 4, 1, 1, 1, 1, 1 $ | $756$ | $4$ | $(3,4,9,8)$ | |
| $ 5, 4 $ | $18144$ | $20$ | $(1,6,5,7,2)(3,4,9,8)$ | |
| $ 4, 2, 1, 1, 1 $ | $7560$ | $4$ | $(1,6,4,2)(5,7)$ | |
| $ 3, 2, 2, 1, 1 $ | $7560$ | $6$ | $(2,7)(3,8,9)(5,6)$ | |
| $ 4, 3, 1, 1 $ | $15120$ | $12$ | $(2,6,7,5)(3,9,8)$ | |
| $ 3, 2, 1, 1, 1, 1 $ | $2520$ | $6$ | $(2,6)(3,9,8)$ | |
| $ 4, 3, 2 $ | $15120$ | $12$ | $(1,4)(2,6,7,5)(3,8,9)$ | |
| $ 7, 1, 1 $ | $25920$ | $7$ | $(2,3,6,8,4,7,9)$ | |
| $ 7, 2 $ | $25920$ | $14$ | $(1,5)(2,8,9,6,7,3,4)$ | |
| $ 9 $ | $40320$ | $9$ | $(1,5,7,4,6,2,9,3,8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $362880=2^{7} \cdot 3^{4} \cdot 5 \cdot 7$ | 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: | 362880.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);