Show commands:
Magma
magma: G := TransitiveGroup(18, 5);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_9$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $18$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,6,8,10,12,14,16,17)(2,3,5,7,9,11,13,15,18), (1,15)(2,16)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: $D_{9}$
Low degree siblings
9T3Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 4, 6, 8,10,12,14,16,17)( 2, 3, 5, 7, 9,11,13,15,18)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 6,10,14,17, 4, 8,12,16)( 2, 5, 9,13,18, 3, 7,11,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3, 9,15)( 4,10,16)( 5,11,18)( 6,12,17)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1,10,17, 8,16, 6,14, 4,12)( 2, 9,18, 7,15, 5,13, 3,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $18=2 \cdot 3^{2}$ | 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: | 18.1 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 . . . . 3 2 . 2 2 2 2 1a 2a 9a 9b 3a 9c 2P 1a 1a 9b 9c 3a 9a 3P 1a 2a 3a 3a 1a 3a 5P 1a 2a 9c 9a 3a 9b 7P 1a 2a 9b 9c 3a 9a X.1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 X.3 2 . -1 -1 2 -1 X.4 2 . A B -1 C X.5 2 . B C -1 A X.6 2 . C A -1 B A = E(9)^2+E(9)^7 B = E(9)^4+E(9)^5 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7 |
magma: CharacterTable(G);