Properties

Label 18T49
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:S_3.C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $49$
Group :  $C_3^2:S_3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,14)(2,9,15)(3,7,13)(4,10,18)(5,11,16)(6,12,17), (1,6,15,10,3,5,14,12,2,4,13,11)(7,17,8,18,9,16)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Low degree siblings

18T49

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 15]
Character table:   
      2  2  2  .  2   2  2   2   2   2   2  2   2  2  .
      3  3  1  2  3   1  3   1   1   1   1  1   1  1  2

        1a 2a 3a 3b  6a 3c  6b 12a 12b 12c 4a 12d 4b 3d
     2P 1a 1a 3a 3c  3c 3b  3b  6b  6b  6a 2a  6a 2a 3d
     3P 1a 2a 1a 1a  2a 1a  2a  4a  4b  4a 4b  4b 4a 1a
     5P 1a 2a 3a 3c  6b 3b  6a 12c 12d 12a 4a 12b 4b 3d
     7P 1a 2a 3a 3b  6a 3c  6b 12b 12a 12d 4b 12c 4a 3d
    11P 1a 2a 3a 3c  6b 3b  6a 12d 12c 12b 4b 12a 4a 3d

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

A = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
B = -E(3)^2
  = (1+Sqrt(-3))/2 = 1+b3
C = -E(4)
  = -Sqrt(-1) = -i
D = -E(12)^11