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Properties

Label 18T19
18T19 1 5 1->5 15 1->15 2 6 2->6 13 2->13 3 4 3->4 14 3->14 9 4->9 12 4->12 7 5->7 10 5->10 8 6->8 11 6->11 7->2 18 7->18 8->3 16 8->16 9->1 17 9->17 10->6 10->14 11->4 11->15 12->5 12->13 13->3 13->17 14->1 14->18 15->2 15->16 16->9 16->10 17->7 17->11 18->8 18->12
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_9$

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Show commands: Magma

magma: G := TransitiveGroup(18, 19);
 

Group invariants

Abstract group:  $C_3\times D_9$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,5,7,2,6,8,3,4,9)(10,14,18,12,13,17,11,15,16)$, $(1,15,2,13,3,14)(4,12,5,10,6,11)(7,18,8,16,9,17)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$, $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: None

Low degree siblings

27T9

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{9}$ $9$ $2$ $9$ $( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
3B $3^{6}$ $2$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3C1 $3^{3},1^{9}$ $2$ $3$ $6$ $(1,3,2)(4,6,5)(7,9,8)$
3C-1 $3^{3},1^{9}$ $2$ $3$ $6$ $(10,11,12)(13,14,15)(16,17,18)$
6A1 $6^{3}$ $9$ $6$ $15$ $( 1,14, 3,13, 2,15)( 4,11, 6,10, 5,12)( 7,17, 9,16, 8,18)$
6A-1 $6^{3}$ $9$ $6$ $15$ $( 1,15, 2,13, 3,14)( 4,12, 5,10, 6,11)( 7,18, 8,16, 9,17)$
9A1 $9^{2}$ $2$ $9$ $16$ $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,17,12,14,16,11,13,18)$
9A2 $9^{2}$ $2$ $9$ $16$ $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,17,14,11,18,15,12,16,13)$
9A4 $9^{2}$ $2$ $9$ $16$ $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,14,18,12,13,17,11,15,16)$
9B1 $9^{2}$ $2$ $9$ $16$ $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,13,16,12,15,18,11,14,17)$
9B-1 $9^{2}$ $2$ $9$ $16$ $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,18,12,13,17,11,15,16)$
9B2 $9^{2}$ $2$ $9$ $16$ $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,16,15,11,17,13,12,18,14)$
9B-2 $9^{2}$ $2$ $9$ $16$ $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,18,13,11,16,14,12,17,15)$
9B4 $9^{2}$ $2$ $9$ $16$ $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,15,17,12,14,16,11,13,18)$
9B-4 $9^{2}$ $2$ $9$ $16$ $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,13,16,12,15,18,11,14,17)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1 9B2 9B-2 9B4 9B-4
Size 1 9 1 1 2 2 2 9 9 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A1 3A-1 9A2 9A4 9A1 9B2 9B-2 9B4 9B-4 9B-1 9B1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 3B 3B 3B 3B 3B 3B 3B 3B 3B
Type
54.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
54.3.1b R 1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
54.3.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
54.3.1c2 C 1 1 ζ3 ζ3−1 ζ3 1 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
54.3.1d1 C 1 −1 ζ3−1 ζ3 ζ3−1 1 ζ3 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
54.3.1d2 C 1 −1 ζ3 ζ3−1 ζ3 1 ζ3−1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
54.3.2a R 2 0 2 2 2 2 2 0 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
54.3.2b1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2 2ζ3 0 0 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3
54.3.2b2 C 2 0 2ζ3 2ζ3−1 2ζ3 2 2ζ3−1 0 0 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1
54.3.2c1 R 2 0 2 2 −1 −1 −1 0 0 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−1+ζ9
54.3.2c2 R 2 0 2 2 −1 −1 −1 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−4+ζ94
54.3.2c3 R 2 0 2 2 −1 −1 −1 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−2+ζ92
54.3.2d1 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ9 ζ9+ζ92 ζ9−2+ζ9−1 −ζ9−4−ζ92+ζ94 ζ9−1+ζ9 ζ9−4−ζ9−ζ94 ζ92+ζ94
54.3.2d2 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ92+ζ94 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4+ζ9 ζ9−1+ζ9 ζ92+ζ94 ζ9−4−ζ9−ζ94
54.3.2d3 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ92+ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−4−ζ9−ζ94 ζ9−4+ζ94 ζ9+ζ92 ζ9−2+ζ9−1
54.3.2d4 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4−ζ9−ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ92+ζ94 ζ9−4+ζ94 ζ9−2+ζ9−1 ζ9+ζ92
54.3.2d5 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ9−1 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ92 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9
54.3.2d6 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9+ζ92 ζ92+ζ94 ζ9−4−ζ9−ζ94 ζ9−2+ζ9−1 ζ9−2+ζ92 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94

magma: CharacterTable(G);
 

Regular extensions

Data not computed