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Magma
magma: G := TransitiveGroup(18, 14);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_9:C_6$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,17,15,13,11,9,8,6,3,2,18,16,14,12,10,7,5,4), (1,3,11,8,10,18,14,15,5)(2,4,12,7,9,17,13,16,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $27$: $C_9:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_9:C_3$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 3,10,15)( 4, 9,16)( 5,18,11)( 6,17,12)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 3,15,10)( 4,16, 9)( 5,11,18)( 6,12,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 9,15, 4,10,16)( 5,17,11, 6,18,12)( 7, 8)(13,14)$ |
| $ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3,16,10, 4,15, 9)( 5,12,18, 6,11,17)( 7, 8)(13,14)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 3, 5, 8,10,11,14,15,18)( 2, 4, 6, 7, 9,12,13,16,17)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 3,11, 8,10,18,14,15, 5)( 2, 4,12, 7, 9,17,13,16, 6)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 3,18, 8,10, 5,14,15,11)( 2, 4,17, 7, 9, 6,13,16,12)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 4, 5, 7,10,12,14,16,18, 2, 3, 6, 8, 9,11,13,15,17)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 4,11, 7,10,17,14,16, 5, 2, 3,12, 8, 9,18,13,15, 6)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 4,18, 7,10, 6,14,16,11, 2, 3,17, 8, 9, 5,13,15,12)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 5,15,14,18,10, 8,11, 3)( 2, 6,16,13,17, 9, 7,12, 4)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 5,10,14,18, 3, 8,11,15)( 2, 6, 9,13,17, 4, 7,12,16)$ |
| $ 9, 9 $ | $3$ | $9$ | $( 1, 5, 3,14,18,15, 8,11,10)( 2, 6, 4,13,17,16, 7,12, 9)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 6,15,13,18, 9, 8,12, 3, 2, 5,16,14,17,10, 7,11, 4)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 6,10,13,18, 4, 8,12,15, 2, 5, 9,14,17, 3, 7,11,16)$ |
| $ 18 $ | $3$ | $18$ | $( 1, 6, 3,13,18,16, 8,12,10, 2, 5, 4,14,17,15, 7,11, 9)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 7,14, 2, 8,13)( 3, 9,15, 4,10,16)( 5,12,18, 6,11,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3,10,15)( 4, 9,16)( 5,11,18)( 6,12,17)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,13, 8, 2,14, 7)( 3,16,10, 4,15, 9)( 5,17,11, 6,18,12)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,14, 8)( 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$ |
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: | $2$ | ||
| Label: | 54.11 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);