Properties

Label 18T21
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 $t$ :  $21$
Group :  $He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,11)(2,6,12)(3,7,13)(4,8,14)(9,15,17)(10,16,18), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18)
$|\Aut(F/K)|$:  $6$

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: $C_3^2 : C_6$

Low degree siblings

9T11, 9T13, 18T20, 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 $ $3$ $3$ $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5, 9, 8)( 6,10, 7)(11,14,15)(12,13,16)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 8,16, 9,13)( 6,11, 7,15,10,14)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,13, 9,16, 8,12)( 6,14,10,15, 7,11)$
$ 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 $ $6$ $3$ $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 5,14)( 2, 6,13)( 3, 7,16)( 4, 8,15)( 9,11,17)(10,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 5,15)( 2, 6,16)( 3, 7,12)( 4, 8,11)( 9,14,17)(10,13,18)$

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   1   1  1  3   2   2  2

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

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

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