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Properties

Label 18T108
Order \(216\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3\times A_4):S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $108$
Group :  $(C_3\times A_4):S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,3,2,5,4)(7,14,10,16,12,18,8,13,9,15,11,17), (1,3,5)(2,4,6)(7,14,12,18,9,15)(8,13,11,17,10,16), (1,12)(2,11)(3,7)(4,8)(5,9)(6,10)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
24:  $S_4$
54:  $(C_3^2:C_3):C_2$
72:  12T44

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

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

Low degree siblings

18T107

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 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 7, 9,12)( 8,10,11)(13,17,16)(14,18,15)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $6$ $6$ $( 7,10,12, 8, 9,11)(13,18,16,14,17,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 7,13)( 8,14)( 9,16)(10,15)(11,18)(12,17)$
$ 6, 3, 3, 2, 2, 2 $ $6$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,12)( 8,10,11)(13,18,16,14,17,15)$
$ 6, 3, 3, 2, 2, 2 $ $6$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,12, 8, 9,11)(13,17,16)(14,18,15)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7,13, 8,14)( 9,16,10,15)(11,18,12,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,17)(14,15,18)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,17,14,16,18)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,13,12,17, 9,16)( 8,14,11,18,10,15)$
$ 12, 6 $ $18$ $12$ $( 1, 4, 5, 2, 3, 6)( 7,13,11,18, 9,16, 8,14,12,17,10,15)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,11, 9, 8,12,10)(13,18,16,14,17,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,17,16)(14,18,15)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,13, 9,16,12,17)( 8,14,10,15,11,18)$
$ 12, 6 $ $18$ $12$ $( 1, 6, 3, 2, 5, 4)( 7,13,10,15,12,17, 8,14, 9,16,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $24$ $3$ $( 1, 7,13)( 2, 8,14)( 3, 9,16)( 4,10,15)( 5,12,17)( 6,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $24$ $3$ $( 1, 7,15)( 2, 8,16)( 3, 9,18)( 4,10,17)( 5,12,14)( 6,11,13)$
$ 3, 3, 3, 3, 3, 3 $ $24$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,12,16)( 6,11,15)$

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 95]
Character table:    
      2  3  3  2  2  2   2   2  2  3   3   2   2   3  3   2   2  .  .  .
      3  3  2  2  2  1   2   2  1  3   2   1   1   2  3   1   1  2  2  2

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

X.1      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  1
X.3      2  2  2  2  .   2   2  .  2   2   .   .   2  2   .   . -1 -1 -1
X.4      2  2 -1 -1  .  -1  -1  .  2   2   .   .   2  2   .   .  2 -1 -1
X.5      2  2 -1 -1  .  -1  -1  .  2   2   .   .   2  2   .   . -1 -1  2
X.6      2  2 -1 -1  .  -1  -1  .  2   2   .   .   2  2   .   . -1  2 -1
X.7      3 -1  3 -1 -1  -1  -1  1  3  -1  -1   1  -1  3  -1   1  .  .  .
X.8      3 -1  3 -1  1  -1  -1 -1  3  -1   1  -1  -1  3   1  -1  .  .  .
X.9      3  3  .  . -1   .   . -1  B   B   D   D  /B /B  /D  /D  .  .  .
X.10     3  3  .  . -1   .   . -1 /B  /B  /D  /D   B  B   D   D  .  .  .
X.11     3  3  .  .  1   .   .  1  B   B  -D  -D  /B /B -/D -/D  .  .  .
X.12     3  3  .  .  1   .   .  1 /B  /B -/D -/D   B  B  -D  -D  .  .  .
X.13     3 -1  .  2 -1   A  /A  1  B   D   D  -D  /D /B  /D -/D  .  .  .
X.14     3 -1  .  2 -1  /A   A  1 /B  /D  /D -/D   D  B   D  -D  .  .  .
X.15     3 -1  .  2  1   A  /A -1  B   D  -D   D  /D /B -/D  /D  .  .  .
X.16     3 -1  .  2  1  /A   A -1 /B  /D -/D  /D   D  B  -D   D  .  .  .
X.17     6 -2 -3  1  .   1   1  .  6  -2   .   .  -2  6   .   .  .  .  .
X.18     6 -2  . -2  .  -A -/A  .  C  -A   .   . -/A /C   .   .  .  .  .
X.19     6 -2  . -2  . -/A  -A  . /C -/A   .   .  -A  C   .   .  .  .  .

A = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 6*E(3)^2
  = -3-3*Sqrt(-3) = -3-3i3
D = -E(3)^2
  = (1+Sqrt(-3))/2 = 1+b3