Properties

Label 18T24
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $24$
Group :  $C_3^2:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (5,7,9)(6,8,10)(11,15,13)(12,16,14), (1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

9T12 x 4, 18T24 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, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 7, 9)( 6, 8,10)(11,15,13)(12,16,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 4,17, 2, 3,18)( 5,11, 9,15, 7,13)( 6,12,10,16, 8,14)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 5, 3, 7,17, 9)( 2, 6, 4, 8,18,10)(11,16,13,12,15,14)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,11)( 2, 5,12)( 3, 8,13)( 4, 7,14)( 9,16,18)(10,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,13)( 2, 5,14)( 3, 8,15)( 4, 7,16)( 9,12,18)(10,11,17)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,15)( 2, 5,16)( 3, 8,11)( 4, 7,12)( 9,14,18)(10,13,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 8]
Character table:   
      2  1  .  1  1   1   1  .  .  .  1
      3  3  2  1  3   1   1  2  2  2  3

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

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

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