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Properties

Label 18T17
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 17);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $S_3$, $C_6$ x 4
$9$:  $C_3^2$
$18$:  $S_3\times C_3$ x 4, $C_6 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$, $S_3\times C_3$ x 3

Degree 9: None

Low degree siblings

18T17 x 3, 27T15

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $54=2 \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  54.12
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);