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Magma
magma: G := TransitiveGroup(18, 46);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times S_3^2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,13,3,7,15)(2,8,14)(4,12,16,6,10,18)(5,11,17), (1,11,9,16,14,6)(2,12,7,17,15,4)(3,10,8,18,13,5), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,18,14,16,15,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$ x 2, $C_6$ x 3 $12$: $D_{6}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ x 2 $36$: $S_3^2$, $C_6\times S_3$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$, $S_3\times C_3$
Degree 9: None
Low degree siblings
12T70, 18T43, 18T46, 27T36, 36T80, 36T82 x 2, 36T92Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 4,10,16)( 5,11,17)( 6,12,18)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 4,16,10)( 5,17,11)( 6,18,12)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)$ |
$ 6, 3, 2, 2, 2, 1, 1, 1 $ | $6$ | $6$ | $( 2, 3)( 4,10,16)( 5,12,17, 6,11,18)( 8, 9)(14,15)$ |
$ 6, 3, 2, 2, 2, 1, 1, 1 $ | $6$ | $6$ | $( 2, 3)( 4,16,10)( 5,18,11, 6,17,12)( 8, 9)(14,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2, 3)( 4,11,18)( 5,12,16)( 6,10,17)( 7, 8, 9)(13,14,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2, 3)( 4,17,12)( 5,18,10)( 6,16,11)( 7, 8, 9)(13,14,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,16)(14,17)(15,18)$ |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 4, 7,10,13,16)( 2, 5, 8,11,14,17)( 3, 6, 9,12,15,18)$ |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 4,13,16, 7,10)( 2, 5,14,17, 8,11)( 3, 6,15,18, 9,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 6)( 3, 5)( 7,10)( 8,12)( 9,11)(13,16)(14,18)(15,17)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1, 4, 7,10,13,16)( 2, 6, 8,12,14,18)( 3, 5, 9,11,15,17)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1, 4,13,16, 7,10)( 2, 6,14,18, 8,12)( 3, 5,15,17, 9,11)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5, 3, 4, 2, 6)( 7,11, 9,10, 8,12)(13,17,15,16,14,18)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5, 9,10,14,18)( 2, 6, 7,11,15,16)( 3, 4, 8,12,13,17)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,15,16, 8,12)( 2, 6,13,17, 9,10)( 3, 4,14,18, 7,11)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,16,10)( 5,17,11)( 6,18,12)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 7,13)( 2, 9,14, 3, 8,15)( 4,10,16)( 5,12,17, 6,11,18)$ |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 7,13)( 2, 9,14, 3, 8,15)( 4,16,10)( 5,18,11, 6,17,12)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,15)( 2, 9,13)( 3, 7,14)( 4,11,18)( 5,12,16)( 6,10,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,15)( 2, 9,13)( 3, 7,14)( 4,17,12)( 5,18,10)( 6,16,11)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,13, 7)( 2,14, 8)( 3,15, 9)( 4,16,10)( 5,17,11)( 6,18,12)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1,13, 7)( 2,15, 8, 3,14, 9)( 4,16,10)( 5,18,11, 6,17,12)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,14, 9)( 2,15, 7)( 3,13, 8)( 4,17,12)( 5,18,10)( 6,16,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | 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: | not nilpotent | ||
Label: | 108.38 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);