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Properties

Label 18T20
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 $t$ :  $20$
Group :  $He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,5,3,15,7)(2,11,6,4,16,8)(9,18,14,10,17,13), (1,2)(3,17)(4,18)(5,10)(6,9)(7,8)(11,16)(12,15)(13,14)
$|\Aut(F/K)|$:  $6$

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: $C_3$

Degree 6: $C_6$

Degree 9: $C_3^2 : S_3 $

Low degree siblings

9T11, 9T13, 18T21, 18T22

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

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  1  3   2  2   1   2  2   1

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

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

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