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Properties

Label 18T19
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 action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_9$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(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

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

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:  54.3
magma: IdentifyGroup(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 9B1 9B-1 9B4 9B-4 9B-2 9B2
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 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
54.3.1c2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
54.3.1d1 C 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
54.3.1d2 C 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
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 2 2ζ3−1 2ζ3 0 0 −1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
54.3.2b2 C 2 0 2ζ3 2ζ3−1 2 2ζ3 2ζ3−1 0 0 −1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
54.3.2c1 R 2 0 2 2 −1 −1 −1 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94
54.3.2c2 R 2 0 2 2 −1 −1 −1 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92
54.3.2c3 R 2 0 2 2 −1 −1 −1 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9
54.3.2d1 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ92+ζ94 ζ9−4−ζ9−ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−2+ζ9−1 ζ9+ζ92
54.3.2d2 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ9−1
54.3.2d3 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94
54.3.2d4 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9+ζ92 ζ9−2+ζ9−1 ζ92+ζ94 ζ9−4−ζ9−ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9
54.3.2d5 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ9−1 ζ92+ζ94 ζ9−4−ζ9−ζ94
54.3.2d6 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94

magma: CharacterTable(G);