Properties

Label 18T22
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $22$
Group :  $He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,8,18,14,5)(2,12,9,17,15,4)(3,11,7,16,13,6), (1,18,9,6,13,12)(2,17,7,5,14,11)(3,16,8,4,15,10)
$|\Aut(F/K)|$:  $3$

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: None

Degree 6: $S_3\times C_3$

Degree 9: None

Low degree siblings

9T11, 9T13, 18T20, 18T21

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 5]
Character table:   
      2  1   .   .  .  1   1   1  1  .  1
      3  3   2   2  3  1   1   1  2  2  2

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

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

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