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Properties

Label 18T4
Degree $18$
Order $18$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2 : C_2$

Related objects

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Group action invariants

Degree $n$:  $18$
Transitive number $t$:  $4$
Group:  $C_3^2 : C_2$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $18$
Generators:  (1,8,14)(2,7,13)(3,10,16)(4,9,15)(5,11,17)(6,12,18), (1,4,17)(2,3,18)(5,8,9)(6,7,10)(11,14,15)(12,13,16), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 6: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

9T5

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

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [18, 4]
Character table:    
     2  1  1  .  .  .  .
     3  2  .  2  2  2  2

       1a 2a 3a 3b 3c 3d
    2P 1a 1a 3a 3b 3c 3d
    3P 1a 2a 1a 1a 1a 1a

X.1     1  1  1  1  1  1
X.2     1 -1  1  1  1  1
X.3     2  .  2 -1 -1 -1
X.4     2  . -1  2 -1 -1
X.5     2  . -1 -1 -1  2
X.6     2  . -1 -1  2 -1