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Properties

Label 18T3
Degree $18$
Order $18$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 3);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $18$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,11)(2,10,12)(3,6,13)(4,5,14)(7,16,18)(8,15,17), (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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

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

Degree 9: $S_3\times C_3$

Low degree siblings

6T5, 9T4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $18=2 \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  18.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1
Size 1 3 1 1 2 2 2 3 3
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A
Type
18.3.1a R 1 1 1 1 1 1 1 1 1
18.3.1b R 1 −1 1 1 1 1 1 −1 −1
18.3.1c1 C 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1
18.3.1c2 C 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3
18.3.1d1 C 1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1
18.3.1d2 C 1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3
18.3.2a R 2 0 2 2 −1 −1 −1 0 0
18.3.2b1 C 2 0 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0
18.3.2b2 C 2 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0

magma: CharacterTable(G);