Show commands:
Magma
magma: G := TransitiveGroup(9, 10);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_9:C_3):C_2$ | ||
CHM label: | $[3^{2}]S(3)_{6}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9), (1,8)(2,7)(3,6)(4,5) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Low degree siblings
18T18, 27T14Siblings 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, 1 $ | $1$ | $1$ | $()$ |
$ 6, 2, 1 $ | $9$ | $6$ | $(2,3,5,9,8,6)(4,7)$ |
$ 3, 3, 1, 1, 1 $ | $3$ | $3$ | $(2,5,8)(3,9,6)$ |
$ 6, 2, 1 $ | $9$ | $6$ | $(2,6,8,9,5,3)(4,7)$ |
$ 3, 3, 1, 1, 1 $ | $3$ | $3$ | $(2,8,5)(3,6,9)$ |
$ 2, 2, 2, 2, 1 $ | $9$ | $2$ | $(2,9)(3,8)(4,7)(5,6)$ |
$ 9 $ | $6$ | $9$ | $(1,2,3,4,5,6,7,8,9)$ |
$ 9 $ | $6$ | $9$ | $(1,2,6,4,5,9,7,8,3)$ |
$ 9 $ | $6$ | $9$ | $(1,2,9,4,5,3,7,8,6)$ |
$ 3, 3, 3 $ | $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $54=2 \cdot 3^{3}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 54.6 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 . . . . 3 3 1 2 1 2 1 2 2 2 3 1a 6a 3a 6b 3b 2a 9a 9b 9c 3c 2P 1a 3a 3b 3b 3a 1a 9a 9c 9b 3c 3P 1a 2a 1a 2a 1a 2a 3c 3c 3c 1a 5P 1a 6b 3b 6a 3a 2a 9a 9c 9b 3c 7P 1a 6a 3a 6b 3b 2a 9a 9b 9c 3c X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 1 1 1 X.3 1 A -/A /A -A -1 1 -/A -A 1 X.4 1 /A -A A -/A -1 1 -A -/A 1 X.5 1 -/A -A -A -/A 1 1 -A -/A 1 X.6 1 -A -/A -/A -A 1 1 -/A -A 1 X.7 2 . 2 . 2 . -1 -1 -1 2 X.8 2 . B . /B . -1 A /A 2 X.9 2 . /B . B . -1 /A A 2 X.10 6 . . . . . . . . -3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |
magma: CharacterTable(G);