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Properties

Label 18T85
Order \(162\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^3:C_6$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $85$
Group :  $C_3^3:C_6$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,18), (1,14,8,4,15,9,17,11,5)(2,13,7,3,16,10,18,12,6)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

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

Low degree siblings

9T22 x 3, 18T85 x 2

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
$ 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 $ $27$ $2$ $( 1, 2)( 3,17)( 4,18)( 5, 6)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$
$ 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 $ $6$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$
$ 3, 3, 3, 3, 3, 3 $ $9$ $3$ $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$
$ 9, 9 $ $18$ $9$ $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$
$ 6, 6, 6 $ $27$ $6$ $( 1, 6,11, 2, 5,12)( 3, 9,13,17, 7,15)( 4,10,14,18, 8,16)$
$ 3, 3, 3, 3, 3, 3 $ $9$ $3$ $( 1,11, 5)( 2,12, 6)( 3,13, 7)( 4,14, 8)( 9,17,15)(10,18,16)$
$ 9, 9 $ $18$ $9$ $( 1,11, 8, 4,14, 9,17,15, 5)( 2,12, 7, 3,13,10,18,16, 6)$
$ 6, 6, 6 $ $27$ $6$ $( 1,12, 5, 2,11, 6)( 3,15, 7,17,13, 9)( 4,16, 8,18,14,10)$

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 11]
Character table:    
      2  1  .  .  .  1  .  .  1   .   1  1   .   1
      3  4  3  3  3  1  4  3  2   2   1  2   2   1

        1a 3a 3b 3c 2a 3d 3e 3f  9a  6a 3g  9b  6b
     2P 1a 3a 3b 3c 1a 3d 3e 3g  9b  3g 3f  9a  3f
     3P 1a 1a 1a 1a 2a 1a 1a 1a  3d  2a 1a  3d  2a
     5P 1a 3a 3b 3c 2a 3d 3e 3g  9b  6b 3f  9a  6a
     7P 1a 3a 3b 3c 2a 3d 3e 3f  9a  6a 3g  9b  6b

X.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
X.3      1  1  1  1 -1  1  1  A   A  -A /A  /A -/A
X.4      1  1  1  1 -1  1  1 /A  /A -/A  A   A  -A
X.5      1  1  1  1  1  1  1  A   A   A /A  /A  /A
X.6      1  1  1  1  1  1  1 /A  /A  /A  A   A   A
X.7      2 -1 -1  2  .  2 -1  2  -1   .  2  -1   .
X.8      2 -1 -1  2  .  2 -1  B  -A   . /B -/A   .
X.9      2 -1 -1  2  .  2 -1 /B -/A   .  B  -A   .
X.10     6  . -3  .  . -3  3  .   .   .  .   .   .
X.11     6  3  .  .  . -3 -3  .   .   .  .   .   .
X.12     6 -3  3  .  . -3  .  .   .   .  .   .   .
X.13     6  .  . -3  .  6  .  .   .   .  .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3