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Magma
magma: G := TransitiveGroup(9, 13);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2 : S_3 $ | ||
CHM label: | $E(9):D_{6}=[3^{2}:2]3=[1/2.S(3)^{2}]3$ | ||
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: | (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9) | 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: $C_3$
Low degree siblings
9T11, 18T20, 18T21, 18T22, 27T11Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1 $ | $6$ | $3$ | $(3,4,5)(6,8,7)$ |
$ 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $(2,9)(4,5)(6,7)$ |
$ 3, 3, 3 $ | $2$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ |
$ 3, 3, 3 $ | $6$ | $3$ | $(1,3,6)(2,4,7)(5,8,9)$ |
$ 3, 3, 3 $ | $3$ | $3$ | $(1,3,8)(2,4,6)(5,7,9)$ |
$ 6, 3 $ | $9$ | $6$ | $(1,3,6,2,5,7)(4,8,9)$ |
$ 3, 3, 3 $ | $6$ | $3$ | $(1,6,3)(2,7,4)(5,9,8)$ |
$ 3, 3, 3 $ | $3$ | $3$ | $(1,6,5)(2,7,3)(4,9,8)$ |
$ 6, 3 $ | $9$ | $6$ | $(1,6,4,9,7,3)(2,8,5)$ |
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: | 54.5 | magma: IdentifyGroup(G);
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Character table: |
2 1 . 1 . . 1 1 . 1 1 3 3 2 1 3 2 2 1 2 2 1 1a 3a 2a 3b 3c 3d 6a 3e 3f 6b 2P 1a 3a 1a 3b 3e 3f 3f 3c 3d 3d 3P 1a 1a 2a 1a 1a 1a 2a 1a 1a 2a 5P 1a 3a 2a 3b 3e 3f 6b 3c 3d 6a 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 1 -1 1 A A -A /A /A -/A X.4 1 1 -1 1 /A /A -/A A A -A X.5 1 1 1 1 A A A /A /A /A X.6 1 1 1 1 /A /A /A A A A X.7 2 -1 . 2 -1 2 . -1 2 . X.8 2 -1 . 2 -/A B . -A /B . X.9 2 -1 . 2 -A /B . -/A B . X.10 6 . . -3 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |
magma: CharacterTable(G);