↑

Properties

Label 18T224
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 $t$ :  $224$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,17)(2,3,18)(5,11,8,14,9,15)(6,12,7,13,10,16), (1,16,8,17,12,5,4,13,9)(2,15,7,18,11,6,3,14,10)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

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

Low degree siblings

9T29, 12T175, 18T219, 18T220, 18T223

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 $ $4$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 9, 8)( 6,10, 7)(11,15,14)(12,16,13)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(11,15,14)(12,16,13)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $54$ $6$ $( 1,17, 4)( 2,18, 3)( 5, 7)( 6, 8)( 9,10)(11,16)(12,15)(13,14)$
$ 3, 3, 3, 3, 3, 3 $ $72$ $3$ $( 1, 7,16)( 2, 8,15)( 3, 5,14)( 4, 6,13)( 9,11,18)(10,12,17)$
$ 9, 9 $ $72$ $9$ $( 1, 7,13, 4, 6,12,17,10,16)( 2, 8,14, 3, 5,11,18, 9,15)$
$ 9, 9 $ $72$ $9$ $( 1, 7,12,17,10,13, 4, 6,16)( 2, 8,11,18, 9,14, 3, 5,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 5,14)( 6,13)( 7,16)( 8,15)( 9,11)(10,12)$
$ 6, 6, 3, 3 $ $36$ $6$ $( 1,17, 4)( 2,18, 3)( 5,15, 9,14, 8,11)( 6,16,10,13, 7,12)$
$ 6, 6, 3, 3 $ $36$ $6$ $( 1, 4,17)( 2, 3,18)( 5,11, 8,14, 9,15)( 6,12, 7,13,10,16)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 5,11, 9,15, 8,14)( 6,12,10,16, 7,13)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $18$ $6$ $( 1, 4,17)( 2, 3,18)( 5,15)( 6,16)( 7,12)( 8,11)( 9,14)(10,13)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $18$ $6$ $( 1,17, 4)( 2,18, 3)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 4, 4, 4, 2, 2, 2 $ $162$ $4$ $( 1, 3)( 2, 4)( 5,11, 7,13)( 6,12, 8,14)( 9,15,10,16)(17,18)$

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 704]
Character table:    
      2  3  1  1  2  1  3  2  .   .   .  2   1   1  1   2   2  2
      3  4  4  4  3  3  1  1  2   2   2  2   2   2  2   2   2  .

        1a 3a 3b 3c 3d 2a 6a 3e  9a  9b 2b  6b  6c 6d  6e  6f 4a
     2P 1a 3b 3a 3c 3d 1a 3c 3e  9b  9a 1a  3b  3a 3d  3c  3c 2a
     3P 1a 1a 1a 1a 1a 2a 2a 1a  3b  3a 2b  2b  2b 2b  2b  2b 4a
     5P 1a 3b 3a 3c 3d 2a 6a 3e  9b  9a 2b  6c  6b 6d  6f  6e 4a
     7P 1a 3a 3b 3c 3d 2a 6a 3e  9a  9b 2b  6b  6c 6d  6e  6f 4a

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

A = -E(3)+2*E(3)^2
  = (-1-3*Sqrt(-3))/2 = -2-3b3
B = -2*E(3)+4*E(3)^2
  = -1-3*Sqrt(-3) = -1-3i3
C = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
D = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
E = -2*E(3)^2
  = 1+Sqrt(-3) = 1+i3