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Properties

Label 18T50
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_9$

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Show commands: Magma

magma: G := TransitiveGroup(18, 50);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_9$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,2,15,3,14)(4,11,5,10,6,12)(7,17,8,16,9,18), (1,4)(2,6)(3,5)(8,9)(10,11)(13,17)(14,16)(15,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$18$:  $D_{9}$
$36$:  $S_3^2$, $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: None

Low degree siblings

27T30, 36T86

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 $ $4$ $3$ $(10,11,12)(13,14,15)(16,17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $27$ $2$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,16)(11,18)(12,17)(13,15)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
$ 9, 9 $ $4$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,17,11,14,18,12,15,16)$
$ 9, 9 $ $4$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,14,16,11,15,17,12,13,18)$
$ 9, 9 $ $2$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,18,11,13,16,12,14,17)$
$ 9, 9 $ $2$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,17,11,14,18,12,15,16)$
$ 9, 9 $ $4$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,14,16,11,15,17,12,13,18)$
$ 9, 9 $ $2$ $9$ $( 1, 6, 7, 2, 4, 8, 3, 5, 9)(10,14,16,11,15,17,12,13,18)$
$ 18 $ $6$ $18$ $( 1,10, 4,15, 9,18, 2,11, 5,13, 7,16, 3,12, 6,14, 8,17)$
$ 18 $ $6$ $18$ $( 1,10, 5,13, 8,17, 2,11, 6,14, 9,18, 3,12, 4,15, 7,16)$
$ 18 $ $6$ $18$ $( 1,10, 6,14, 7,16, 2,11, 4,15, 8,17, 3,12, 5,13, 9,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1,10)( 2,12)( 3,11)( 4,17)( 5,16)( 6,18)( 7,13)( 8,15)( 9,14)$
$ 6, 6, 6 $ $18$ $6$ $( 1,10, 2,12, 3,11)( 4,17, 5,16, 6,18)( 7,13, 8,15, 9,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,10)( 8,11)( 9,12)$
$ 6, 6, 6 $ $6$ $6$ $( 1,13, 2,14, 3,15)( 4,16, 5,17, 6,18)( 7,10, 8,11, 9,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.16
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 6A 6B 9A1 9A2 9A4 9B1 9B2 9B4 18A1 18A5 18A7
Size 1 3 9 27 2 2 4 6 18 2 2 2 4 4 4 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 3A 3B 9A2 9A4 9A1 9B2 9B4 9B1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2B 3A 3A 3A 3A 3A 3A 6A 6A 6A
Type
108.16.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.16.1b R 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1
108.16.1c R 1 −1 1 −1 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1
108.16.1d R 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 1 1 1 1 1
108.16.2a R 2 0 2 0 2 −1 −1 0 −1 2 2 2 −1 −1 −1 0 0 0
108.16.2b R 2 2 0 0 2 2 2 2 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
108.16.2c R 2 −2 0 0 2 2 2 −2 0 −1 −1 −1 −1 −1 −1 1 1 1
108.16.2d R 2 0 −2 0 2 −1 −1 0 1 2 2 2 −1 −1 −1 0 0 0
108.16.2e1 R 2 2 0 0 −1 2 −1 −1 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
108.16.2e2 R 2 2 0 0 −1 2 −1 −1 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
108.16.2e3 R 2 2 0 0 −1 2 −1 −1 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
108.16.2f1 R 2 −2 0 0 −1 2 −1 1 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
108.16.2f2 R 2 −2 0 0 −1 2 −1 1 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
108.16.2f3 R 2 −2 0 0 −1 2 −1 1 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9
108.16.4a R 4 0 0 0 4 −2 −2 0 0 −2 −2 −2 1 1 1 0 0 0
108.16.4b1 R 4 0 0 0 −2 −2 1 0 0 2ζ9−4+2ζ94 2ζ9−1+2ζ9 2ζ9−2+2ζ92 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94 0 0 0
108.16.4b2 R 4 0 0 0 −2 −2 1 0 0 2ζ9−2+2ζ92 2ζ9−4+2ζ94 2ζ9−1+2ζ9 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92 0 0 0
108.16.4b3 R 4 0 0 0 −2 −2 1 0 0 2ζ9−1+2ζ9 2ζ9−2+2ζ92 2ζ9−4+2ζ94 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9 0 0 0

magma: CharacterTable(G);