Show commands:
Magma
magma: G := TransitiveGroup(9, 1);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_9$ | ||
CHM label: | $C(9)=9$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (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) $3$: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
$ 9 $ | $1$ | $9$ | $(1,2,3,4,5,6,7,8,9)$ |
$ 9 $ | $1$ | $9$ | $(1,3,5,7,9,2,4,6,8)$ |
$ 3, 3, 3 $ | $1$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ |
$ 9 $ | $1$ | $9$ | $(1,5,9,4,8,3,7,2,6)$ |
$ 9 $ | $1$ | $9$ | $(1,6,2,7,3,8,4,9,5)$ |
$ 3, 3, 3 $ | $1$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ |
$ 9 $ | $1$ | $9$ | $(1,8,6,4,2,9,7,5,3)$ |
$ 9 $ | $1$ | $9$ | $(1,9,8,7,6,5,4,3,2)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $9=3^{2}$ | magma: Order(G);
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Cyclic: | yes | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 9.1 | magma: IdentifyGroup(G);
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Character table: |
3 2 2 2 2 2 2 2 2 2 1a 9a 9b 3a 9c 9d 3b 9e 9f X.1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 A /A 1 A /A X.3 1 /A A 1 /A A 1 /A A X.4 1 B /D A C /C /A D /B X.5 1 C /B A D /D /A B /C X.6 1 D /C A B /B /A C /D X.7 1 /D C /A /B B A /C D X.8 1 /C B /A /D D A /B C X.9 1 /B D /A /C C A /D B A = E(3) = (-1+Sqrt(-3))/2 = b3 B = -E(9)^4-E(9)^7 C = E(9)^4 D = E(9)^7 |
magma: CharacterTable(G);