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Properties

Label 18T24
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 24);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (5,7,9)(6,8,10)(11,15,13)(12,16,14), (1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

9T12 x 4, 18T24 x 3, 27T6

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  54.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 3D 3E 6A1 6A-1
Size 1 9 1 1 6 6 6 6 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 3D 3E 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A
Type
54.8.1a R 1 1 1 1 1 1 1 1 1 1
54.8.1b R 1 −1 1 1 1 1 1 1 −1 −1
54.8.2a R 2 0 2 2 −1 −1 −1 2 0 0
54.8.2b R 2 0 2 2 −1 −1 2 −1 0 0
54.8.2c R 2 0 2 2 −1 2 −1 −1 0 0
54.8.2d R 2 0 2 2 2 −1 −1 −1 0 0
54.8.3a1 C 3 1 3ζ3−1 3ζ3 0 0 0 0 ζ3 ζ3−1
54.8.3a2 C 3 1 3ζ3 3ζ3−1 0 0 0 0 ζ3−1 ζ3
54.8.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1
54.8.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3

magma: CharacterTable(G);