Properties

Label 18T8
Order \(36\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4 \times C_3$

Related objects

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4$

Degree 9: $C_3^2$

Low degree siblings

12T20 x 3

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 11]
Character table:   
      2  2  2  2   2   2  2  .  .  .  .  .  .
      3  2  1  2   1   1  2  2  2  2  2  2  2

        1a 2a 3a  6a  6b 3b 3c 3d 3e 3f 3g 3h
     2P 1a 1a 3b  3b  3a 3a 3h 3g 3f 3e 3d 3c
     3P 1a 2a 1a  2a  2a 1a 1a 1a 1a 1a 1a 1a
     5P 1a 2a 3b  6b  6a 3a 3h 3g 3f 3e 3d 3c

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

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