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Magma
magma: G := TransitiveGroup(9, 24);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $((C_3^3:C_3):C_2):C_2$ | ||
CHM label: | $[3^{3}:2]S(3)$ | ||
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: | (1,2,9), (3,6)(4,7)(5,8), (1,2)(4,5)(7,8), (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) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $12$: $D_{6}$ x 2 $36$: $S_3^2$ $108$: $C_3^2 : D_{6} $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Low degree siblings
9T24 x 2, 18T129 x 3, 18T136 x 3, 18T137 x 3, 27T121, 27T128 x 3, 27T129, 36T502 x 3Siblings 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, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(6,7,8)$ |
$ 3, 3, 1, 1, 1 $ | $6$ | $3$ | $(3,4,5)(6,7,8)$ |
$ 3, 3, 1, 1, 1 $ | $6$ | $3$ | $(3,4,5)(6,8,7)$ |
$ 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $(3,6)(4,7)(5,8)$ |
$ 6, 1, 1, 1 $ | $18$ | $6$ | $(3,6,4,7,5,8)$ |
$ 2, 2, 2, 1, 1, 1 $ | $27$ | $2$ | $(2,9)(4,5)(7,8)$ |
$ 2, 2, 2, 2, 1 $ | $27$ | $2$ | $(2,9)(3,6)(4,8)(5,7)$ |
$ 6, 2, 1 $ | $54$ | $6$ | $(2,9)(3,6,4,8,5,7)$ |
$ 3, 3, 3 $ | $2$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ |
$ 3, 3, 3 $ | $6$ | $3$ | $(1,2,9)(3,4,5)(6,8,7)$ |
$ 3, 2, 2, 2 $ | $18$ | $6$ | $(1,2,9)(3,6)(4,7)(5,8)$ |
$ 6, 3 $ | $18$ | $6$ | $(1,2,9)(3,6,4,7,5,8)$ |
$ 6, 3 $ | $18$ | $6$ | $(1,2,9)(3,6,5,8,4,7)$ |
$ 3, 3, 3 $ | $18$ | $3$ | $(1,3,6)(2,4,7)(5,8,9)$ |
$ 9 $ | $36$ | $9$ | $(1,3,6,2,4,7,9,5,8)$ |
$ 6, 3 $ | $54$ | $6$ | $(1,3,6)(2,5,7,9,4,8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $324=2^{2} \cdot 3^{4}$ | 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: | 324.39 | magma: IdentifyGroup(G);
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Character table: |
2 2 1 1 1 2 1 2 2 1 1 1 1 1 1 1 . 1 3 4 3 3 3 2 2 1 1 1 4 3 2 2 2 2 2 1 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f 2P 1a 3a 3b 3c 1a 3b 1a 1a 3c 3d 3e 3a 3e 3d 3f 9a 3f 3P 1a 1a 1a 1a 2a 2a 2b 2c 2c 1a 1a 2a 2a 2a 1a 3d 2b 5P 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f 7P 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 X.3 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 X.5 2 2 2 2 . . -2 . . 2 2 . . . -1 -1 1 X.6 2 2 2 2 . . 2 . . 2 2 . . . -1 -1 -1 X.7 2 -1 -1 2 -2 1 . . . 2 -1 1 1 -2 2 -1 . X.8 2 -1 -1 2 2 -1 . . . 2 -1 -1 -1 2 2 -1 . X.9 4 -2 -2 4 . . . . . 4 -2 . . . -2 1 . X.10 6 -3 3 . -2 1 . . . -3 . 1 -2 1 . . . X.11 6 -3 3 . 2 -1 . . . -3 . -1 2 -1 . . . X.12 6 . -3 . -2 1 . . . -3 3 -2 1 1 . . . X.13 6 . -3 . 2 -1 . . . -3 3 2 -1 -1 . . . X.14 6 . . -3 . . . -2 1 6 . . . . . . . X.15 6 . . -3 . . . 2 -1 6 . . . . . . . X.16 6 3 . . -2 -2 . . . -3 -3 1 1 1 . . . X.17 6 3 . . 2 2 . . . -3 -3 -1 -1 -1 . . . |
magma: CharacterTable(G);