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Properties

Label 18T6
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 6);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_6$
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:  (1,15,8,4,13,9)(2,16,7,3,14,10)(5,17,12)(6,18,11), (1,3,18,2,4,17)(5,8,10,6,7,9)(11,14,15,12,13,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

Low degree siblings

12T18, 18T6, 36T6

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

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:  36.12
magma: IdentifyGroup(G);
 
Character table:    
      2  2  2  2  2  1  1   1   2   1   2   2   2   1   2   1   2   2   2
      3  2  1  2  1  2  2   2   1   2   1   2   2   2   1   2   1   2   2

        1a 2a 2b 2c 6a 3a  6b  6c  3b  6d  6e  3c  3d  6f  6g  6h  3e  6i
     2P 1a 1a 1a 1a 3a 3a  3d  3e  3d  3e  3e  3e  3b  3c  3b  3c  3c  3c
     3P 1a 2a 2b 2c 2b 1a  2b  2c  1a  2a  2b  1a  1a  2a  2b  2c  1a  2b
     5P 1a 2a 2b 2c 6a 3a  6g  6h  3d  6f  6i  3e  3b  6d  6b  6c  3c  6e

X.1      1  1  1  1  1  1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1 -1 -1  1 -1  1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1
X.3      1 -1  1 -1  1  1   1  -1   1  -1   1   1   1  -1   1  -1   1   1
X.4      1  1 -1 -1 -1  1  -1  -1   1   1  -1   1   1   1  -1  -1   1  -1
X.5      1 -1 -1  1 -1  1   A  -A  -A   A   A  -A -/A  /A  /A -/A -/A  /A
X.6      1 -1 -1  1 -1  1  /A -/A -/A  /A  /A -/A  -A   A   A  -A  -A   A
X.7      1 -1  1 -1  1  1 -/A  /A -/A  /A -/A -/A  -A   A  -A   A  -A  -A
X.8      1 -1  1 -1  1  1  -A   A  -A   A  -A  -A -/A  /A -/A  /A -/A -/A
X.9      1  1 -1 -1 -1  1   A   A  -A  -A   A  -A -/A -/A  /A  /A -/A  /A
X.10     1  1 -1 -1 -1  1  /A  /A -/A -/A  /A -/A  -A  -A   A   A  -A   A
X.11     1  1  1  1  1  1 -/A -/A -/A -/A -/A -/A  -A  -A  -A  -A  -A  -A
X.12     1  1  1  1  1  1  -A  -A  -A  -A  -A  -A -/A -/A -/A -/A -/A -/A
X.13     2  . -2  .  1 -1   1   .  -1   .  -2   2  -1   .   1   .   2  -2
X.14     2  .  2  . -1 -1  -1   .  -1   .   2   2  -1   .  -1   .   2   2
X.15     2  . -2  .  1 -1 -/A   .  /A   .   B  -B   A   .  -A   . -/B  /B
X.16     2  . -2  .  1 -1  -A   .   A   .  /B -/B  /A   . -/A   .  -B   B
X.17     2  .  2  . -1 -1  /A   .  /A   .  -B  -B   A   .   A   . -/B -/B
X.18     2  .  2  . -1 -1   A   .   A   . -/B -/B  /A   .  /A   .  -B  -B

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = -2*E(3)^2
  = 1+Sqrt(-3) = 1+i3

magma: CharacterTable(G);