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Properties

Label 18T20
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 20);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,5,3,15,7)(2,11,6,4,16,8)(9,18,14,10,17,13), (1,2)(3,17)(4,18)(5,10)(6,9)(7,8)(11,16)(12,15)(13,14)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_3^2 : S_3 $

Low degree siblings

9T11, 9T13, 18T21, 18T22, 27T11

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  54.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B1 3B-1 3C 3D1 3D-1 6A1 6A-1
Size 1 9 2 3 3 6 6 6 9 9
2 P 1A 1A 3A 3B-1 3B1 3C 3D-1 3D1 3B1 3B-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A
Type
54.5.1a R 1 1 1 1 1 1 1 1 1 1
54.5.1b R 1 −1 1 1 1 1 1 1 −1 −1
54.5.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1
54.5.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3
54.5.1d1 C 1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1
54.5.1d2 C 1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3
54.5.2a R 2 0 2 2 2 −1 −1 −1 0 0
54.5.2b1 C 2 0 2 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0
54.5.2b2 C 2 0 2 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0
54.5.6a R 6 0 −3 0 0 0 0 0 0 0

magma: CharacterTable(G);