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Properties

Label 18T11
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, 11);
 

Group action invariants

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

Degree 9: $S_3^2$

Low degree siblings

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

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:    
     2  2  2  2  2  1  1  .  1  1
     3  2  1  1  .  2  1  2  1  2

       1a 2a 2b 2c 3a 6a 3b 6b 3c
    2P 1a 1a 1a 1a 3a 3a 3b 3c 3c
    3P 1a 2a 2b 2c 1a 2b 1a 2a 1a
    5P 1a 2a 2b 2c 3a 6a 3b 6b 3c

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1 -1  1  1 -1  1 -1  1
X.3     1 -1  1 -1  1  1  1 -1  1
X.4     1  1 -1 -1  1 -1  1  1  1
X.5     2 -2  .  .  2  . -1  1 -1
X.6     2  2  .  .  2  . -1 -1 -1
X.7     2  . -2  . -1  1 -1  .  2
X.8     2  .  2  . -1 -1 -1  .  2
X.9     4  .  .  . -2  .  1  . -2

magma: CharacterTable(G);