↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma

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

Group invariants

Abstract group:  $S_3\times D_9$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{9}$ $3$ $2$ $9$ $( 1,15)( 2,13)( 3,14)( 4,18)( 5,16)( 6,17)( 7,12)( 8,10)( 9,11)$
2B $2^{9}$ $9$ $2$ $9$ $( 1,14)( 2,13)( 3,15)( 4,12)( 5,11)( 6,10)( 7,18)( 8,17)( 9,16)$
2C $2^{8},1^{2}$ $27$ $2$ $8$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,16)(11,18)(12,17)(13,15)$
3A $3^{6}$ $2$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
3B $3^{6}$ $2$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3C $3^{3},1^{9}$ $4$ $3$ $6$ $(1,3,2)(4,6,5)(7,9,8)$
6A $6^{3}$ $6$ $6$ $15$ $( 1,13, 3,15, 2,14)( 4,16, 6,18, 5,17)( 7,10, 9,12, 8,11)$
6B $6^{3}$ $18$ $6$ $15$ $( 1,15, 2,14, 3,13)( 4,10, 5,12, 6,11)( 7,16, 8,18, 9,17)$
9A1 $9^{2}$ $2$ $9$ $16$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,18,11,13,16,12,14,17)$
9A2 $9^{2}$ $2$ $9$ $16$ $( 1, 6, 7, 2, 4, 8, 3, 5, 9)(10,14,16,11,15,17,12,13,18)$
9A4 $9^{2}$ $2$ $9$ $16$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,17,11,14,18,12,15,16)$
9B1 $9^{2}$ $4$ $9$ $16$ $( 1, 6, 7, 2, 4, 8, 3, 5, 9)(10,15,18,11,13,16,12,14,17)$
9B2 $9^{2}$ $4$ $9$ $16$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,14,16,11,15,17,12,13,18)$
9B4 $9^{2}$ $4$ $9$ $16$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,17,11,14,18,12,15,16)$
18A1 $18$ $6$ $18$ $17$ $( 1,18, 9,13, 5,12, 3,17, 8,15, 4,11, 2,16, 7,14, 6,10)$
18A5 $18$ $6$ $18$ $17$ $( 1,16, 8,13, 6,11, 3,18, 7,15, 5,10, 2,17, 9,14, 4,12)$
18A7 $18$ $6$ $18$ $17$ $( 1,17, 7,13, 4,10, 3,16, 9,15, 6,12, 2,18, 8,14, 5,11)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(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 9A2 9A1 9A4
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);
 

Regular extensions

Data not computed