Properties

Label 18T483
Degree $18$
Order $5184$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3^3:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 483);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $483$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3^3:S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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);
 

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 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 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);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5184.br
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);