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Magma
magma: G := TransitiveGroup(9, 25);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ | ||
CHM label: | $[1/2.S(3)^{3}]3$ | ||
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), (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) $3$: $C_3$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Low degree siblings
12T132 x 2, 12T133, 18T141 x 2, 18T142, 18T143, 27T130, 27T131, 36T511, 36T512, 36T524, 36T546 x 2, 36T547Siblings 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 $ | $6$ | $3$ | $(6,7,8)$ |
$ 2, 2, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $(4,5)(7,8)$ |
$ 3, 3, 1, 1, 1 $ | $12$ | $3$ | $(3,4,5)(6,7,8)$ |
$ 3, 2, 2, 1, 1 $ | $54$ | $6$ | $(2,9)(4,5)(6,7,8)$ |
$ 3, 3, 3 $ | $4$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ |
$ 3, 3, 3 $ | $4$ | $3$ | $(1,2,9)(3,4,5)(6,8,7)$ |
$ 3, 3, 3 $ | $36$ | $3$ | $(1,3,6)(2,4,7)(5,8,9)$ |
$ 9 $ | $36$ | $9$ | $(1,3,6,2,4,7,9,5,8)$ |
$ 9 $ | $36$ | $9$ | $(1,3,6,9,5,8,2,4,7)$ |
$ 3, 3, 3 $ | $36$ | $3$ | $(1,6,3)(2,7,4)(5,9,8)$ |
$ 9 $ | $36$ | $9$ | $(1,6,4,2,7,5,9,8,3)$ |
$ 9 $ | $36$ | $9$ | $(1,6,5,9,8,4,2,7,3)$ |
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: | 324.160 | magma: IdentifyGroup(G);
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Character table: |
2 2 1 2 . 1 . . . . . . . . 3 4 3 1 3 1 4 4 2 2 2 2 2 2 1a 3a 2a 3b 6a 3c 3d 3e 9a 9b 3f 9c 9d 2P 1a 3a 1a 3b 3a 3d 3c 3f 9d 9c 3e 9b 9a 3P 1a 1a 2a 1a 2a 1a 1a 1a 3c 3d 1a 3c 3d 5P 1a 3a 2a 3b 6a 3d 3c 3f 9d 9c 3e 9b 9a 7P 1a 3a 2a 3b 6a 3c 3d 3e 9a 9b 3f 9c 9d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 B B B /B /B /B X.3 1 1 1 1 1 1 1 /B /B /B B B B X.4 3 3 -1 3 -1 3 3 . . . . . . X.5 4 -2 . 1 . A /A B 1 /B /B B 1 X.6 4 -2 . 1 . /A A /B 1 B B /B 1 X.7 4 -2 . 1 . A /A /B B 1 B 1 /B X.8 4 -2 . 1 . /A A B /B 1 /B 1 B X.9 4 -2 . 1 . A /A 1 /B B 1 /B B X.10 4 -2 . 1 . /A A 1 B /B 1 B /B X.11 6 3 -2 . 1 -3 -3 . . . . . . X.12 6 3 2 . -1 -3 -3 . . . . . . X.13 12 . . -3 . 3 3 . . . . . . A = -E(3)+2*E(3)^2 = (-1-3*Sqrt(-3))/2 = -2-3b3 B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |
magma: CharacterTable(G);