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Magma
magma: G := TransitiveGroup(18, 15);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times \He_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,7,12,2,8,11)(3,9,14,4,10,13)(5,16,18,6,15,17), (1,2)(3,4)(5,9,7,6,10,8)(11,13,15,12,14,16)(17,18) | 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 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $27$: $C_3^2:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_3^2:C_3$
Low degree siblings
18T15 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5, 7,10)( 6, 8, 9)(11,15,14)(12,16,13)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5,10, 7)( 6, 9, 8)(11,14,15)(12,13,16)$ |
| $ 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 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 8,10, 6, 7, 9)(11,16,14,12,15,13)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 9, 7, 6,10, 8)(11,13,15,12,14,16)(17,18)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 8,10, 6, 7, 9)(11,13,15,12,14,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 7,10)( 6, 8, 9)(11,14,15)(12,13,16)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5,12, 2, 6,11)( 3, 8,14, 4, 7,13)( 9,15,17,10,16,18)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5,13, 2, 6,14)( 3, 8,15, 4, 7,16)( 9,11,17,10,12,18)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5,16, 2, 6,15)( 3, 8,11, 4, 7,12)( 9,14,17,10,13,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 6,12)( 2, 5,11)( 3, 7,14)( 4, 8,13)( 9,16,17)(10,15,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 6,13)( 2, 5,14)( 3, 7,15)( 4, 8,16)( 9,12,17)(10,11,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 6,16)( 2, 5,15)( 3, 7,11)( 4, 8,12)( 9,13,17)(10,14,18)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1,11, 6, 2,12, 5)( 3,13, 7, 4,14, 8)( 9,18,16,10,17,15)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1,11, 9, 2,12,10)( 3,13, 5, 4,14, 6)( 7,17,15, 8,18,16)$ |
| $ 6, 6, 6 $ | $3$ | $6$ | $( 1,11, 8, 2,12, 7)( 3,13,10, 4,14, 9)( 5,17,15, 6,18,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,12, 6)( 2,11, 5)( 3,14, 7)( 4,13, 8)( 9,17,16)(10,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,12, 9)( 2,11,10)( 3,14, 5)( 4,13, 6)( 7,18,15)( 8,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,12, 8)( 2,11, 7)( 3,14,10)( 4,13, 9)( 5,18,15)( 6,17,16)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17, 4)( 2,18, 3)( 5,10, 7)( 6, 9, 8)(11,15,14)(12,16,13)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,18, 4, 2,17, 3)( 5, 9, 7, 6,10, 8)(11,16,14,12,15,13)$ |
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: | 54.10 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);