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Magma
magma: G := TransitiveGroup(18, 286);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $286$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3^3:C_6$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,16,8,17,14,9)(2,15,7,18,13,10)(3,12,6,4,11,5), (1,4,18,2,3,17)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $A_4$, $C_6\times C_2$ $24$: $A_4\times C_2$ x 3 $48$: $C_2^2 \times A_4$ $648$: $S_3 \wr C_3 $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $S_3 \wr C_3 $
Low degree siblings
18T283 x 4, 18T285 x 2, 18T286, 24T2876 x 2, 36T2078 x 4, 36T2079 x 4, 36T2080 x 4, 36T2081 x 4, 36T2093, 36T2094, 36T2095 x 2, 36T2097 x 2, 36T2098 x 2, 36T2099 x 2, 36T2100 x 2, 36T2101 x 2, 36T2102 x 2Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 7, 9)( 6, 8,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 1, 3,18)( 2, 4,17)( 5, 7, 9)( 6, 8,10)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3,18)( 2, 4,17)( 5, 7, 9)( 6, 8,10)(11,14,15)(12,13,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,18)( 4,17)(13,16)(14,15)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 3,18)( 4,17)( 5, 7, 9)( 6, 8,10)(13,16)(14,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3, 6,11)( 4, 5,12)( 9,16,17)(10,15,18)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,10,15,18, 6,11, 3, 8,14)( 2, 9,16,17, 5,12, 4, 7,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $36$ | $3$ | $( 1,14, 8)( 2,13, 7)( 3,11, 6)( 4,12, 5)( 9,17,16)(10,18,15)$ |
| $ 9, 9 $ | $72$ | $9$ | $( 1,14,10,18,15, 6, 3,11, 8)( 2,13, 9,17,16, 5, 4,12, 7)$ |
| $ 6, 6, 6 $ | $108$ | $6$ | $( 1,16, 8,17,14, 9)( 2,15, 7,18,13,10)( 3,12, 6, 4,11, 5)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1,17)( 2,18)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,12)(13,15)(14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 4,18, 2, 3,17)( 5, 6)( 7,10)( 8, 9)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1, 4,18, 2, 3,17)( 5, 6)( 7,10)( 8, 9)(11,13,15,12,14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,13,15,12,14,16)(17,18)$ |
| $ 6, 6, 6 $ | $108$ | $6$ | $( 1, 9,14,17, 8,16)( 2,10,13,18, 7,15)( 3, 5,11, 4, 6,12)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(13,16)(14,15)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 5, 7, 9)( 6, 8,10)(13,16)(14,15)$ |
| $ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 1, 3,18)( 2, 4,17)(13,16)(14,15)$ |
| $ 3, 3, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 3,18)( 2, 4,17)( 5, 7, 9)( 6, 8,10)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3,18)( 4,17)( 7, 9)( 8,10)(13,16)(14,15)$ |
| $ 6, 6, 3, 3 $ | $108$ | $6$ | $( 1, 8,14,18,10,15)( 2, 7,13,17, 9,16)( 3, 6,11)( 4, 5,12)$ |
| $ 6, 6, 3, 3 $ | $108$ | $6$ | $( 1,14,10,18,15, 8)( 2,13, 9,17,16, 7)( 3,11, 6)( 4,12, 5)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1,16,10, 2,15, 9)( 3,12, 6, 4,11, 5)( 7,18,13, 8,17,14)$ |
| $ 18 $ | $72$ | $18$ | $( 1,16, 6, 4,11, 7,18,13,10, 2,15, 5, 3,12, 8,17,14, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1,17)( 2,18)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,12)(13,14)(15,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1,17)( 2,18)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,13,15,12,14,16)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 4,18, 2, 3,17)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 6, 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 4,18, 2, 3,17)( 5, 8, 9, 6, 7,10)(11,12)(13,14)(15,16)$ |
| $ 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, 6 $ | $8$ | $6$ | $( 1, 4,18, 2, 3,17)( 5, 8, 9, 6, 7,10)(11,13,15,12,14,16)$ |
| $ 6, 6, 6 $ | $36$ | $6$ | $( 1, 9,14, 2,10,13)( 3, 5,11, 4, 6,12)( 7,15,17, 8,16,18)$ |
| $ 18 $ | $72$ | $18$ | $( 1, 5,11, 4, 8,16,18, 9,14, 2, 6,12, 3, 7,15,17,10,13)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1296=2^{4} \cdot 3^{4}$ | 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: | 1296.3494 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);