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Properties

Label 18T18
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_9:C_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $18$
Group :  $D_9:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10), (1,15,7,9,13,3)(2,16,8,10,14,4)(5,6)(11,18)(12,17)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_9:C_3):C_2$

Low degree siblings

9T10

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 6]
Character table:    
      2  1  1  1   1   1  1  .   .   .  .
      3  3  2  2   1   1  1  2   2   2  3

        1a 3a 3b  6a  6b 2a 9a  9b  9c 3c
     2P 1a 3b 3a  3a  3b 1a 9a  9c  9b 3c
     3P 1a 1a 1a  2a  2a 2a 3c  3c  3c 1a
     5P 1a 3b 3a  6b  6a 2a 9a  9c  9b 3c
     7P 1a 3a 3b  6a  6b 2a 9a  9b  9c 3c

X.1      1  1  1   1   1  1  1   1   1  1
X.2      1  1  1  -1  -1 -1  1   1   1  1
X.3      1  A /A -/A  -A -1  1   A  /A  1
X.4      1 /A  A  -A -/A -1  1  /A   A  1
X.5      1  A /A  /A   A  1  1   A  /A  1
X.6      1 /A  A   A  /A  1  1  /A   A  1
X.7      2  2  2   .   .  . -1  -1  -1  2
X.8      2  B /B   .   .  . -1 -/A  -A  2
X.9      2 /B  B   .   .  . -1  -A -/A  2
X.10     6  .  .   .   .  .  .   .   . -3

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