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Properties

Label 18T1
Degree $18$
Order $18$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(18, 1);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_9$

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $18=2 \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  18.2
magma: IdentifyGroup(G);
 
Character table:    
      2  1  1  1   1   1  1  1   1   1  1  1   1   1  1   1  1   1  1
      3  2  2  2   2   2  2  2   2   2  2  2   2   2  2   2  2   2  2

        1a 2a 9a 18a 18b 9b 3a  6a 18c 9c 9d 18d  6b 3b 18e 9e 18f 9f

X.1      1  1  1   1   1  1  1   1   1  1  1   1   1  1   1  1   1  1
X.2      1 -1  1  -1  -1  1  1  -1  -1  1  1  -1  -1  1  -1  1  -1  1
X.3      1  1  A   A  /A /A  1   1   A  A /A  /A   1  1   A  A  /A /A
X.4      1 -1  A  -A -/A /A  1  -1  -A  A /A -/A  -1  1  -A  A -/A /A
X.5      1  1 /A  /A   A  A  1   1  /A /A  A   A   1  1  /A /A   A  A
X.6      1 -1 /A -/A  -A  A  1  -1 -/A /A  A  -A  -1  1 -/A /A  -A  A
X.7      1  1  B   B  /D /D  A   A   C  C /C  /C  /A /A   D  D  /B /B
X.8      1 -1  B  -B -/D /D  A  -A  -C  C /C -/C -/A /A  -D  D -/B /B
X.9      1  1  C   C  /B /B  A   A   D  D /D  /D  /A /A   B  B  /C /C
X.10     1 -1  C  -C -/B /B  A  -A  -D  D /D -/D -/A /A  -B  B -/C /C
X.11     1  1  D   D  /C /C  A   A   B  B /B  /B  /A /A   C  C  /D /D
X.12     1 -1  D  -D -/C /C  A  -A  -B  B /B -/B -/A /A  -C  C -/D /D
X.13     1  1 /D  /D   C  C /A  /A  /B /B  B   B   A  A  /C /C   D  D
X.14     1 -1 /D -/D  -C  C /A -/A -/B /B  B  -B  -A  A -/C /C  -D  D
X.15     1  1 /C  /C   B  B /A  /A  /D /D  D   D   A  A  /B /B   C  C
X.16     1 -1 /C -/C  -B  B /A -/A -/D /D  D  -D  -A  A -/B /B  -C  C
X.17     1  1 /B  /B   D  D /A  /A  /C /C  C   C   A  A  /D /D   B  B
X.18     1 -1 /B -/B  -D  D /A -/A -/C /C  C  -C  -A  A -/D /D  -B  B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(9)^2
C = -E(9)^2-E(9)^5
D = E(9)^5

magma: CharacterTable(G);