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Properties

Label 18T43
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times S_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(18, 43);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times S_3^2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,13,4,8,18)(2,10,14,5,9,16)(3,11,15,6,7,17), (1,8,2,9,3,7)(4,16,5,17,6,18)(10,12,11)(13,15,14)
magma: Generators(G);
 

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: $C_3$

Degree 6: $C_6$, $S_3^2$

Degree 9: None

Low degree siblings

12T70, 18T46 x 2, 27T36, 36T80, 36T82 x 2, 36T92

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7,13)( 8,14)( 9,15)(10,17)(11,18)(12,16)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4,10,17)( 5,11,18)( 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 6, 6, 3, 3 $ $9$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,14, 9,13, 8,15)(10,18,12,17,11,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7, 8, 9)(13,14,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
$ 6, 6, 3, 3 $ $9$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,15, 8,13, 9,14)(10,16,11,17,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7, 9, 8)(13,15,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,11)( 8,12)( 9,10)(13,18)(14,16)(15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,18)( 8,16)( 9,17)(10,15)(11,13)(12,14)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 4, 9,10,15,17)( 2, 5, 7,11,13,18)( 3, 6, 8,12,14,16)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 4, 9,17,15,10)( 2, 5, 7,18,13,11)( 3, 6, 8,16,14,12)$
$ 6, 6, 6 $ $3$ $6$ $( 1, 5, 3, 4, 2, 6)( 7,12, 9,11, 8,10)(13,16,15,18,14,17)$
$ 6, 6, 6 $ $3$ $6$ $( 1, 5, 3, 4, 2, 6)( 7,16, 9,18, 8,17)(10,13,12,15,11,14)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5, 8,10,13,16)( 2, 6, 9,11,14,17)( 3, 4, 7,12,15,18)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5, 8,17,13,12)( 2, 6, 9,18,14,10)( 3, 4, 7,16,15,11)$
$ 6, 6, 6 $ $3$ $6$ $( 1, 6, 2, 4, 3, 5)( 7,10, 8,11, 9,12)(13,17,14,18,15,16)$
$ 6, 6, 6 $ $3$ $6$ $( 1, 6, 2, 4, 3, 5)( 7,17, 8,18, 9,16)(10,14,11,15,12,13)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 6, 7,10,14,18)( 2, 4, 8,11,15,16)( 3, 5, 9,12,13,17)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 6, 7,17,14,11)( 2, 4, 8,18,15,12)( 3, 5, 9,16,13,10)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,11,16)( 5,12,17)( 6,10,18)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,18,12)( 5,16,10)( 6,17,11)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 8,13)( 2, 9,14)( 3, 7,15)( 4,12,18)( 5,10,16)( 6,11,17)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 8,13)( 2, 9,14)( 3, 7,15)( 4,16,11)( 5,17,12)( 6,18,10)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4,10,17)( 5,11,18)( 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4,17,10)( 5,18,11)( 6,16,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.38
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);