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Properties

Label 18T9
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(18, 9);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3^2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5,4,8,17,9)(2,6,3,7,18,10)(11,16,14,12,15,13), (3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11), (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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 6: $D_{6}$ x 2, $S_3^2$

Degree 9: $S_3^2$

Low degree siblings

6T9, 9T8, 12T16, 18T11 x 2, 36T13

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 6A 6B
Size 1 3 3 9 2 2 4 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 3A 3B
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2B
Type
36.10.1a R 1 1 1 1 1 1 1 1 1
36.10.1b R 1 −1 −1 1 1 1 1 −1 −1
36.10.1c R 1 −1 1 −1 1 1 1 −1 1
36.10.1d R 1 1 −1 −1 1 1 1 1 −1
36.10.2a R 2 0 2 0 2 −1 −1 0 −1
36.10.2b R 2 2 0 0 −1 2 −1 −1 0
36.10.2c R 2 −2 0 0 −1 2 −1 1 0
36.10.2d R 2 0 −2 0 2 −1 −1 0 1
36.10.4a R 4 0 0 0 −2 −2 1 0 0

magma: CharacterTable(G);