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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

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{9}$ $1$ $2$ $9$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1,14, 7)( 2,13, 8)( 3,16,10)( 4,15, 9)( 5,17,12)( 6,18,11)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 7,14)( 2, 8,13)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)$
6A1 $6^{3}$ $1$ $6$ $15$ $( 1, 8,14, 2, 7,13)( 3, 9,16, 4,10,15)( 5,11,17, 6,12,18)$
6A-1 $6^{3}$ $1$ $6$ $15$ $( 1,13, 7, 2,14, 8)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$
9A1 $9^{2}$ $1$ $9$ $16$ $( 1,10,18, 7,16, 6,14, 3,11)( 2, 9,17, 8,15, 5,13, 4,12)$
9A-1 $9^{2}$ $1$ $9$ $16$ $( 1,18,16,14,11,10, 7, 6, 3)( 2,17,15,13,12, 9, 8, 5, 4)$
9A2 $9^{2}$ $1$ $9$ $16$ $( 1,16,11, 7, 3,18,14,10, 6)( 2,15,12, 8, 4,17,13, 9, 5)$
9A-2 $9^{2}$ $1$ $9$ $16$ $( 1, 3, 6, 7,10,11,14,16,18)( 2, 4, 5, 8, 9,12,13,15,17)$
9A4 $9^{2}$ $1$ $9$ $16$ $( 1, 6,10,14,18, 3, 7,11,16)( 2, 5, 9,13,17, 4, 8,12,15)$
9A-4 $9^{2}$ $1$ $9$ $16$ $( 1,11, 3,14, 6,16, 7,18,10)( 2,12, 4,13, 5,15, 8,17, 9)$
18A1 $18$ $1$ $18$ $17$ $( 1, 9,18, 8,16, 5,14, 4,11, 2,10,17, 7,15, 6,13, 3,12)$
18A-1 $18$ $1$ $18$ $17$ $( 1,12, 3,13, 6,15, 7,17,10, 2,11, 4,14, 5,16, 8,18, 9)$
18A5 $18$ $1$ $18$ $17$ $( 1,17,16,13,11, 9, 7, 5, 3, 2,18,15,14,12,10, 8, 6, 4)$
18A-5 $18$ $1$ $18$ $17$ $( 1, 4, 6, 8,10,12,14,15,18, 2, 3, 5, 7, 9,11,13,16,17)$
18A7 $18$ $1$ $18$ $17$ $( 1,15,11, 8, 3,17,14, 9, 6, 2,16,12, 7, 4,18,13,10, 5)$
18A-7 $18$ $1$ $18$ $17$ $( 1, 5,10,13,18, 4, 7,12,16, 2, 6, 9,14,17, 3, 8,11,15)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/9$

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:

1A 2A 3A1 3A-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 3A1 3A-1 9A-2 9A-4 9A1 9A4 9A-1 9A2 9A-2 9A2 9A-4 9A4 9A1 9A-1
3 P 1A 2A 1A 1A 2A 2A 3A-1 3A1 3A-1 3A-1 3A1 3A1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
18.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
18.2.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
18.2.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
18.2.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
18.2.1d1 C 1 −1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
18.2.1d2 C 1 −1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
18.2.1e1 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4
18.2.1e2 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94
18.2.1e3 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92
18.2.1e4 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2
18.2.1e5 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1
18.2.1e6 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9
18.2.1f1 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4
18.2.1f2 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94
18.2.1f3 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92
18.2.1f4 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2
18.2.1f5 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1
18.2.1f6 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9

magma: CharacterTable(G);