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Properties

Label 18T21
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 21);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_6$
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:  (1,5,11)(2,6,12)(3,7,13)(4,8,14)(9,15,17)(10,16,18), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $C_3^2 : C_6$

Low degree siblings

9T11, 9T13, 18T20, 18T22, 27T11

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

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:  54.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B1 3B-1 3C 3D1 3D-1 6A1 6A-1
Size 1 9 2 3 3 6 6 6 9 9
2 P 1A 1A 3A 3B-1 3B1 3C 3D-1 3D1 3B1 3B-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A
Type
54.5.1a R 1 1 1 1 1 1 1 1 1 1
54.5.1b R 1 −1 1 1 1 1 1 1 −1 −1
54.5.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1
54.5.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3
54.5.1d1 C 1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1
54.5.1d2 C 1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3
54.5.2a R 2 0 2 2 2 −1 −1 −1 0 0
54.5.2b1 C 2 0 2 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0
54.5.2b2 C 2 0 2 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0
54.5.6a R 6 0 −3 0 0 0 0 0 0 0

magma: CharacterTable(G);