↑

Properties

Label 18T207
Order \(648\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $207$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,14,4,6,15,17,10,11)(2,8,13,3,5,16,18,9,12), (1,17,4)(2,18,3)(5,10)(6,9)(7,8)(11,15,14)(12,16,13)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 9: $S_3 \wr C_3 $

Low degree siblings

9T28, 12T176, 18T197 x 2, 18T198 x 2, 18T202, 18T204, 18T206

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, 3, 3 $ $8$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 9, 8)( 6,10, 7)(11,14,15)(12,13,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 9, 8)( 6,10, 7)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 1,17, 4)( 2,18, 3)(11,15,14)(12,16,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 7, 9)( 8,10)(17,18)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $54$ $6$ $( 1,18)( 2,17)( 3, 4)( 5,10)( 6, 9)( 7, 8)(11,14,15)(12,13,16)$
$ 9, 9 $ $72$ $9$ $( 1, 7,14, 4, 6,15,17,10,11)( 2, 8,13, 3, 5,16,18, 9,12)$
$ 3, 3, 3, 3, 3, 3 $ $36$ $3$ $( 1,10,11)( 2, 9,12)( 3, 8,13)( 4, 7,14)( 5,16,18)( 6,15,17)$
$ 9, 9 $ $72$ $9$ $( 1,14, 6,17,11, 7, 4,15,10)( 2,13, 5,18,12, 8, 3,16, 9)$
$ 3, 3, 3, 3, 3, 3 $ $36$ $3$ $( 1,14,10)( 2,13, 9)( 3,16, 8)( 4,15, 7)( 5,18,12)( 6,17,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(11,13)(12,14)(15,16)$
$ 3, 3, 3, 3, 2, 2, 2 $ $36$ $6$ $( 1, 4,17)( 2, 3,18)( 5, 9, 8)( 6,10, 7)(11,16)(12,15)(13,14)$
$ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $6$ $( 5, 9, 8)( 6,10, 7)(11,13)(12,14)(15,16)$
$ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $6$ $( 1,17, 4)( 2,18, 3)(11,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)(17,18)$
$ 6, 6, 6 $ $108$ $6$ $( 1, 7,14, 2, 8,13)( 3, 5,16,17,10,11)( 4, 6,15,18, 9,12)$
$ 6, 6, 6 $ $108$ $6$ $( 1,14, 8, 3,16,10)( 2,13, 7, 4,15, 9)( 5,18,12, 6,17,11)$

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 705]
Character table:    
      2  3  .  2  1  3  2   .  1   .  1  3  1  2  2  3   1   1
      3  4  4  3  3  1  1   2  2   2  2  2  2  2  2  1   1   1

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

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

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