Properties

Label 18T32
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times A_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $32$
Group :  $S_3\times A_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,14)(2,11,13)(3,7,16)(4,8,15)(5,9,17)(6,10,18), (1,5)(2,6)(7,10)(8,9)(11,12)(13,15)(14,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $A_4\times C_2$

Degree 9: $S_3\times C_3$

Low degree siblings

12T43, 18T31

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 44]
Character table:   
      2  3  3  3  3  2  2   .   1  1   .   1  1
      3  2  1  .  1  2  1   2   1  2   2   1  2

        1a 2a 2b 2c 3a 6a  3b  6b 3c  3d  6c 3e
     2P 1a 1a 1a 1a 3a 3a  3d  3e 3e  3b  3c 3c
     3P 1a 2a 2b 2c 1a 2a  1a  2c 1a  1a  2c 1a
     5P 1a 2a 2b 2c 3a 6a  3d  6c 3e  3b  6b 3c

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

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