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Magma
magma: G := TransitiveGroup(18, 483);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $483$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3^3:S_4$ | ||
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: | (7,8)(9,12)(10,11)(13,16,17,14,15,18), (1,9,17,5,11,13,3,7,15)(2,10,18,6,12,14,4,8,16), (1,6,3,2,5,4)(7,14,12,15,9,18,8,13,11,16,10,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ x 3 $48$: $S_4\times C_2$ x 3 $96$: $V_4^2:S_3$ $192$: 12T100 $1296$: $S_3\wr S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4$
Degree 9: $S_3\wr S_3$
Low degree siblings
18T483, 18T484 x 2, 18T486 x 2, 18T488 x 2, 36T5913 x 2, 36T5914 x 2, 36T5924 x 2, 36T5925 x 2, 36T5945, 36T5948, 36T5955, 36T5956, 36T5957, 36T5958, 36T5963 x 2, 36T5964 x 2, 36T5973 x 2, 36T5974 x 2, 36T5978, 36T5979, 36T5983, 36T5990, 36T5991, 36T5998, 36T6002, 36T6003, 36T6028 x 2, 36T6029 x 2, 36T6030 x 2, 36T6031 x 2, 36T6035, 36T6036, 36T6037 x 2, 36T6038 x 2, 36T6041 x 2, 36T6042 x 2, 36T6238 x 2, 36T6239 x 2, 36T6240 x 2, 36T6241 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 $ | $6$ | $3$ | $(13,17,15)(14,18,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
$ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 7, 8)( 9,12)(10,11)(13,16,17,14,15,18)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
$ 6, 3, 3, 2, 2, 2 $ | $72$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)( 9,12)(10,11)(13,16,17,14,15,18)$ |
$ 3, 3, 2, 2, 2, 2, 2, 2 $ | $36$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 9,11)(10,12)(15,17)(16,18)$ |
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 9,11)(10,12)(15,17)(16,18)$ |
$ 6, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
$ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,10)(11,12)$ |
$ 6, 3, 3, 2, 2, 2 $ | $24$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7, 8)( 9,10)(11,12)(13,17,15)(14,18,16)$ |
$ 6, 6, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,10,11, 8, 9,12)$ |
$ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,10,11, 8, 9,12)(13,17,15)(14,18,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
$ 3, 3, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,17,15)(14,18,16)$ |
$ 6, 6, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 9,11)(10,12)(13,16,17,14,15,18)$ |
$ 6, 2, 2, 2, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(3,5)(4,6)$ |
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $6$ | $( 3, 5)( 4, 6)(13,17,15)(14,18,16)$ |
$ 3, 3, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 3, 5)( 4, 6)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3, 5)( 4, 6)( 9,11)(10,12)(15,17)(16,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $81$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 6)( 2, 5)( 3, 4)(13,16)(14,15)(17,18)$ |
$ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7, 9,11)( 8,10,12)(13,16)(14,15)(17,18)$ |
$ 9, 9 $ | $576$ | $9$ | $( 1, 9,17, 5,11,13, 3, 7,15)( 2,10,18, 6,12,14, 4, 8,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $288$ | $3$ | $( 1, 9,15)( 2,10,16)( 3, 7,13)( 4, 8,14)( 5,11,17)( 6,12,18)$ |
$ 6, 6, 3, 3 $ | $864$ | $6$ | $( 1,12,15)( 2,11,16)( 3, 8,17, 5,10,13)( 4, 7,18, 6, 9,14)$ |
$ 4, 4, 2, 2, 2, 2, 1, 1 $ | $324$ | $4$ | $( 1,12)( 2,11)( 3,10, 5, 8)( 4, 9, 6, 7)(15,17)(16,18)$ |
$ 12, 2, 2, 2 $ | $216$ | $12$ | $( 1, 9, 6, 8, 3,11, 2,10, 5, 7, 4,12)(13,16)(14,15)(17,18)$ |
$ 4, 4, 4, 2, 2, 2 $ | $108$ | $4$ | $( 1,11, 2,12)( 3, 7, 4, 8)( 5, 9, 6,10)(13,16)(14,15)(17,18)$ |
$ 6, 6, 1, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)$ |
$ 6, 6, 3, 3 $ | $144$ | $6$ | $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)(13,17,15)(14,18,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 1,12)( 2,11)( 3, 8)( 4, 7)( 5,10)( 6, 9)$ |
$ 3, 3, 2, 2, 2, 2, 2, 2 $ | $72$ | $6$ | $( 1,12)( 2,11)( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,17,15)(14,18,16)$ |
$ 6, 4, 4, 4 $ | $216$ | $12$ | $( 1,11, 2,12)( 3, 9, 6, 8)( 4,10, 5, 7)(13,16,17,14,15,18)$ |
$ 4, 4, 4, 2, 2, 2 $ | $108$ | $4$ | $( 1,11, 2,12)( 3, 9, 6, 8)( 4,10, 5, 7)(13,14)(15,16)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $108$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(15,17)(16,18)$ |
$ 6, 6, 2, 2, 1, 1 $ | $216$ | $6$ | $( 1, 8, 5,10, 3,12)( 2, 7, 6, 9, 4,11)(15,17)(16,18)$ |
$ 4, 4, 4, 2, 2, 2 $ | $324$ | $4$ | $( 1, 9, 4,12)( 2,10, 3,11)( 5, 7, 6, 8)(13,16)(14,15)(17,18)$ |
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $4$ | $( 1,10, 3,12)( 2, 9, 4,11)( 5, 8)( 6, 7)$ |
$ 4, 4, 3, 3, 2, 2 $ | $216$ | $12$ | $( 1,10, 3,12)( 2, 9, 4,11)( 5, 8)( 6, 7)(13,17,15)(14,18,16)$ |
$ 6, 4, 4, 4 $ | $72$ | $12$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,16,17,14,15,18)$ |
$ 4, 4, 4, 2, 2, 2 $ | $36$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,14)(15,16)(17,18)$ |
$ 12, 6 $ | $144$ | $12$ | $( 1, 7, 6,10, 3,11, 2, 8, 5, 9, 4,12)(13,16,17,14,15,18)$ |
$ 12, 2, 2, 2 $ | $72$ | $12$ | $( 1, 7, 6,10, 3,11, 2, 8, 5, 9, 4,12)(13,14)(15,16)(17,18)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $5184=2^{6} \cdot 3^{4}$ | 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: | 5184.br | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);