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Properties

Label 18T2
Degree $18$
Order $18$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_6 \times C_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 2);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6 \times C_3$
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:  (1,7,13,2,8,14)(3,9,15,4,10,16)(5,11,17,6,12,18), (1,16,5,2,15,6)(3,11,8,4,12,7)(9,17,14,10,18,13)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$ x 4

Degree 6: $C_6$ x 4

Degree 9: $C_3^2$

Low degree siblings

There are no siblings 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,14,16)(12,13,15)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 4,17, 2, 3,18)( 5, 7,10, 6, 8, 9)(11,13,16,12,14,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 5,15)( 2, 6,16)( 3, 8,12)( 4, 7,11)( 9,14,18)(10,13,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 6,15, 2, 5,16)( 3, 7,12, 4, 8,11)( 9,13,18,10,14,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 7,13, 2, 8,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 9,12, 2,10,11)( 3, 6,13, 4, 5,14)( 7,15,18, 8,16,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,10,12)( 2, 9,11)( 3, 5,13)( 4, 6,14)( 7,16,18)( 8,15,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1,11,10, 2,12, 9)( 3,14, 5, 4,13, 6)( 7,17,16, 8,18,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,12,10)( 2,11, 9)( 3,13, 5)( 4,14, 6)( 7,18,16)( 8,17,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,13, 8)( 2,14, 7)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$
$ 6, 6, 6 $ $1$ $6$ $( 1,14, 8, 2,13, 7)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,15, 5)( 2,16, 6)( 3,12, 8)( 4,11, 7)( 9,18,14)(10,17,13)$
$ 6, 6, 6 $ $1$ $6$ $( 1,16, 5, 2,15, 6)( 3,11, 8, 4,12, 7)( 9,17,14,10,18,13)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 3)( 2,18, 4)( 5,10, 8)( 6, 9, 7)(11,16,14)(12,15,13)$
$ 6, 6, 6 $ $1$ $6$ $( 1,18, 3, 2,17, 4)( 5, 9, 8, 6,10, 7)(11,15,14,12,16,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $18=2 \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  18.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A1 3A-1 3D1 3C1 3D-1 3B1 3C-1 3B-1 3A1 3B-1 3D-1 3A-1 3D1 3B1 3C1 3C-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A
Type
18.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
18.5.1b R 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
18.5.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1
18.5.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1
18.5.1d1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3
18.5.1d2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1
18.5.1e1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
18.5.1e2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3
18.5.1f1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
18.5.1f2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
18.5.1g1 C 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −1
18.5.1g2 C 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −1
18.5.1h1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3
18.5.1h2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1
18.5.1i1 C 1 −1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
18.5.1i2 C 1 −1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
18.5.1j1 C 1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
18.5.1j2 C 1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1

magma: CharacterTable(G);