Properties

Label 18T30
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $30$
Group :  $C_3\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5)(2,4,6)(7,18,11,15,9,13,8,17,12,16,10,14), (1,8,18)(2,7,17)(3,10,13)(4,9,14)(5,11,16)(6,12,15)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $S_4$

Degree 9: $S_3\times C_3$

Low degree siblings

12T45, 18T33

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 42]
Character table:   
      2  3  3  2  2  3   3   2   2   3  3   2   2   .   .  .
      3  2  1  1  1  2   1   1   1   1  2   1   1   2   2  2

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

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

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