Properties

Label 18T23
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $23$
Group :  $C_3\times C_3:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,7,6,14,10)(2,17,8,5,15,12)(3,16,9,4,13,11), (1,8,13)(2,9,14)(3,7,15)(4,5,6)(10,11,12)(16,17,18), (1,9,15)(2,7,13)(3,8,14)(4,6,5)(10,12,11)(16,18,17)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$ x 4, $C_6$
18:  $S_3\times C_3$ x 4, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$, $S_3\times C_3$ x 3

Degree 9: None

Low degree siblings

18T23 x 3

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 13]
Character table:   
      2  1   .   .  .   .   .   .   .  1   1   1  1  .   .  .  .   .  1
      3  3   3   3  3   3   3   3   3  1   1   1  3  3   3  3  3   3  3

        1a  3a  3b 3c  3d  3e  3f  3g 2a  6a  6b 3h 3i  3j 3k 3l  3m 3n
     2P 1a  3b  3a 3c  3g  3f  3e  3d 1a  3h  3n 3n 3i  3m 3k 3l  3j 3h
     3P 1a  1a  1a 1a  1a  1a  1a  1a 2a  2a  2a 1a 1a  1a 1a 1a  1a 1a
     5P 1a  3b  3a 3c  3g  3f  3e  3d 2a  6b  6a 3n 3i  3m 3k 3l  3j 3h

X.1      1   1   1  1   1   1   1   1  1   1   1  1  1   1  1  1   1  1
X.2      1   1   1  1   1   1   1   1 -1  -1  -1  1  1   1  1  1   1  1
X.3      1   A  /A  1   A  /A   A  /A -1  -A -/A /A  1  /A  1  1   A  A
X.4      1  /A   A  1  /A   A  /A   A -1 -/A  -A  A  1   A  1  1  /A /A
X.5      1   A  /A  1   A  /A   A  /A  1   A  /A /A  1  /A  1  1   A  A
X.6      1  /A   A  1  /A   A  /A   A  1  /A   A  A  1   A  1  1  /A /A
X.7      2  -1  -1  2  -1  -1  -1  -1  .   .   .  2 -1   2 -1 -1   2  2
X.8      2   2   2 -1  -1  -1  -1  -1  .   .   .  2  2  -1 -1 -1  -1  2
X.9      2  -1  -1 -1  -1   2   2  -1  .   .   .  2 -1  -1 -1  2  -1  2
X.10     2  -1  -1 -1   2  -1  -1   2  .   .   .  2 -1  -1  2 -1  -1  2
X.11     2 -/A  -A  2 -/A  -A -/A  -A  .   .   . /B -1  /B -1 -1   B  B
X.12     2  -A -/A  2  -A -/A  -A -/A  .   .   .  B -1   B -1 -1  /B /B
X.13     2   B  /B -1 -/A  -A -/A  -A  .   .   . /B  2  -A -1 -1 -/A  B
X.14     2  /B   B -1  -A -/A  -A -/A  .   .   .  B  2 -/A -1 -1  -A /B
X.15     2 -/A  -A -1 -/A  /B   B  -A  .   .   . /B -1  -A -1  2 -/A  B
X.16     2  -A -/A -1  -A   B  /B -/A  .   .   .  B -1 -/A -1  2  -A /B
X.17     2 -/A  -A -1   B  -A -/A  /B  .   .   . /B -1  -A  2 -1 -/A  B
X.18     2  -A -/A -1  /B -/A  -A   B  .   .   .  B -1 -/A  2 -1  -A /B

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