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Properties

Label 18T37
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 37);
 

Group invariants

Abstract group:  $C_3:S_4$
magma: IdentifyGroup(G);
 
Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6,3,2,5,4)(7,12,10,8,11,9)(13,18,16)(14,17,15)$, $(1,6)(2,5)(3,4)(7,16,8,15)(9,14,10,13)(11,18,12,17)$, $(1,16)(2,15)(3,13)(4,14)(5,18)(6,17)(7,11)(8,12)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 6: $S_4$

Degree 9: $C_3^2:C_2$

Low degree siblings

12T44 x 3, 18T40, 24T79 x 3, 36T23, 36T56

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 $2^{6},1^{6}$ $3$ $2$ $6$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2B $2^{8},1^{2}$ $18$ $2$ $8$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,17)( 6,18)( 7,11)( 8,12)$
3A $3^{6}$ $2$ $3$ $12$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)$
3B $3^{6}$ $8$ $3$ $12$ $( 1,15,10)( 2,16, 9)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$
3C $3^{6}$ $8$ $3$ $12$ $( 1,14,11)( 2,13,12)( 3,15, 7)( 4,16, 8)( 5,17,10)( 6,18, 9)$
3D $3^{6}$ $8$ $3$ $12$ $( 1, 7,17)( 2, 8,18)( 3,10,14)( 4, 9,13)( 5,11,15)( 6,12,16)$
4A $4^{3},2^{3}$ $18$ $4$ $12$ $( 1,16, 2,15)( 3,13, 4,14)( 5,18, 6,17)( 7,12)( 8,11)( 9,10)$
6A $6^{2},3^{2}$ $6$ $6$ $14$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10, 8,11, 9)(13,17,16,14,18,15)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 3B 3C 3D 4A 6A
Size 1 3 18 2 8 8 8 18 6
2 P 1A 1A 1A 3A 3B 3C 3D 2A 3A
3 P 1A 2A 2B 1A 1A 1A 1A 4A 2A
Type
72.43.1a R 1 1 1 1 1 1 1 1 1
72.43.1b R 1 1 −1 1 1 1 1 −1 1
72.43.2a R 2 2 0 −1 −1 −1 2 0 −1
72.43.2b R 2 2 0 −1 −1 2 −1 0 −1
72.43.2c R 2 2 0 −1 2 −1 −1 0 −1
72.43.2d R 2 2 0 2 −1 −1 −1 0 2
72.43.3a R 3 −1 1 3 0 0 0 −1 −1
72.43.3b R 3 −1 −1 3 0 0 0 1 −1
72.43.6a R 6 −2 0 −3 0 0 0 0 1

magma: CharacterTable(G);
 

Regular extensions

Data not computed