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Properties

Label 18T44
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:S_3.C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $44$
Group :  $C_3\times C_3:S_3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,13,12,3,17,15,11,2,16,14,10)(4,8,5,9,6,7), (4,17,10)(5,18,11)(6,16,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$, $C_3^2:C_4$

Degree 9: None

Low degree siblings

12T73 x 2, 18T44

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 36]
Character table:    
      2  2  2  .  2   2  .  2   2  .   2   2   2   2  2  2  .  .  .
      3  3  1  3  3   1  3  3   1  3   1   1   1   1  1  1  3  3  3

        1a 2a 3a 3b  6a 3c 3d  6b 3e 12a 12b 12c 12d 4a 4b 3f 3g 3h
     2P 1a 1a 3a 3d  3d 3e 3b  3b 3c  6b  6b  6a  6a 2a 2a 3h 3g 3f
     3P 1a 2a 1a 1a  2a 1a 1a  2a 1a  4b  4a  4b  4a 4b 4a 1a 1a 1a
     5P 1a 2a 3a 3d  6b 3e 3b  6a 3c 12c 12d 12a 12b 4a 4b 3h 3g 3f
     7P 1a 2a 3a 3b  6a 3c 3d  6b 3e 12b 12a 12d 12c 4b 4a 3f 3g 3h
    11P 1a 2a 3a 3d  6b 3e 3b  6a 3c 12d 12c 12b 12a 4b 4a 3h 3g 3f

X.1      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  1
X.3      1 -1  1  1  -1  1  1  -1  1   D  -D   D  -D  D -D  1  1  1
X.4      1 -1  1  1  -1  1  1  -1  1  -D   D  -D   D -D  D  1  1  1
X.5      1 -1  1  A  -A  A /A -/A /A   E  -E -/E  /E  D -D /A  1  A
X.6      1 -1  1  A  -A  A /A -/A /A  -E   E  /E -/E -D  D /A  1  A
X.7      1 -1  1 /A -/A /A  A  -A  A -/E  /E   E  -E  D -D  A  1 /A
X.8      1 -1  1 /A -/A /A  A  -A  A  /E -/E  -E   E -D  D  A  1 /A
X.9      1  1  1  A   A  A /A  /A /A  -A  -A -/A -/A -1 -1 /A  1  A
X.10     1  1  1 /A  /A /A  A   A  A -/A -/A  -A  -A -1 -1  A  1 /A
X.11     1  1  1  A   A  A /A  /A /A   A   A  /A  /A  1  1 /A  1  A
X.12     1  1  1 /A  /A /A  A   A  A  /A  /A   A   A  1  1  A  1 /A
X.13     4  . -2  4   . -2  4   . -2   .   .   .   .  .  .  1  1  1
X.14     4  .  1  4   .  1  4   .  1   .   .   .   .  .  . -2 -2 -2
X.15     4  . -2  B   .  C /B   . /C   .   .   .   .  .  . /A  1  A
X.16     4  . -2 /B   . /C  B   .  C   .   .   .   .  .  .  A  1 /A
X.17     4  .  1  B   .  A /B   . /A   .   .   .   .  .  . /C -2  C
X.18     4  .  1 /B   . /A  B   .  A   .   .   .   .  .  .  C -2 /C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
C = -2*E(3)^2
  = 1+Sqrt(-3) = 1+i3
D = -E(4)
  = -Sqrt(-1) = -i
E = -E(12)^11