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Properties

Label 18T3
Degree $18$
Order $18$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_3$

Related objects

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Group action invariants

Degree $n$:  $18$
Transitive number $t$:  $3$
Group:  $S_3 \times C_3$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $18$
Generators:  (1,9,11)(2,10,12)(3,6,13)(4,5,14)(7,16,18)(8,15,17), (1,2)(3,17)(4,18)(5,10)(6,9)(7,8)(11,16)(12,15)(13,14)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $S_3$, $S_3\times C_3$

Degree 9: $S_3\times C_3$

Low degree siblings

6T5, 9T4

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

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [18, 3]
Character table:    
     2  1  1  .   .   1  1   .   1  1
     3  2  1  2   2   1  2   2   1  2

       1a 2a 3a  3b  6a 3c  3d  6b 3e
    2P 1a 1a 3a  3d  3e 3e  3b  3c 3c
    3P 1a 2a 1a  1a  2a 1a  1a  2a 1a
    5P 1a 2a 3a  3d  6b 3e  3b  6a 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   A  -A  A  /A -/A /A
X.4     1 -1  1  /A -/A /A   A  -A  A
X.5     1  1  1   A   A  A  /A  /A /A
X.6     1  1  1  /A  /A /A   A   A  A
X.7     2  . -1  -1   .  2  -1   .  2
X.8     2  . -1 -/A   .  B  -A   . /B
X.9     2  . -1  -A   . /B -/A   .  B

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