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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 action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(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

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  72.43
magma: IdentifyGroup(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);