Properties

Label 18T47
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2.A_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $47$
Group :  $C_3^2.A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,9,4,15,11,5,18,8)(2,14,10,3,16,12,6,17,7), (1,17,9,4,14,11,5,16,8)(2,18,10,3,13,12,6,15,7)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$
27:  $C_9:C_3$
36:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 9: $C_9:C_3$

Low degree siblings

18T47 x 2

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 21]
Character table:   
      2  2  2   2  2   2  2   2   2   2   2   2  2   2  2  .  .  .  .  .  .
      3  3  2   2  2   2  2   2   2   2   2   2  3   2  3  2  2  2  2  2  2

        1a 2a  6a 3a  6b 3b  6c  6d  6e  6f  6g 3c  6h 3d 9a 9b 9c 9d 9e 9f
     2P 1a 1a  3b 3b  3a 3a  3b  3b  3a  3a  3d 3d  3c 3c 9f 9e 9d 9c 9b 9a
     3P 1a 2a  2a 1a  2a 1a  2a  2a  2a  2a  2a 1a  2a 1a 3d 3d 3d 3c 3c 3c
     5P 1a 2a  6b 3b  6a 3a  6e  6f  6c  6d  6h 3d  6g 3c 9f 9e 9d 9c 9b 9a
     7P 1a 2a  6a 3a  6b 3b  6c  6d  6e  6f  6g 3c  6h 3d 9a 9b 9c 9d 9e 9f

X.1      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  A  A  A /A /A /A
X.3      1  1   1  1   1  1   1   1   1   1   1  1   1  1 /A /A /A  A  A  A
X.4      1  1   A  A  /A /A   A   A  /A  /A   1  1   1  1  1  A /A  A /A  1
X.5      1  1  /A /A   A  A  /A  /A   A   A   1  1   1  1  1 /A  A /A  A  1
X.6      1  1   A  A  /A /A   A   A  /A  /A   1  1   1  1  A /A  1  1  A /A
X.7      1  1  /A /A   A  A  /A  /A   A   A   1  1   1  1 /A  A  1  1 /A  A
X.8      1  1   A  A  /A /A   A   A  /A  /A   1  1   1  1 /A  1  A /A  1  A
X.9      1  1  /A /A   A  A  /A  /A   A   A   1  1   1  1  A  1 /A  A  1 /A
X.10     3 -1  -1  3  -1  3  -1  -1  -1  -1  -1  3  -1  3  .  .  .  .  .  .
X.11     3  3   .  .   .  .   .   .   .   .  /C /C   C  C  .  .  .  .  .  .
X.12     3  3   .  .   .  .   .   .   .   .   C  C  /C /C  .  .  .  .  .  .
X.13     3 -1 -/A  C  -A /C -/A -/A  -A  -A  -1  3  -1  3  .  .  .  .  .  .
X.14     3 -1  -A /C -/A  C  -A  -A -/A -/A  -1  3  -1  3  .  .  .  .  .  .
X.15     3 -1   B  .  /B  .  /B   2   B   2  -A /C -/A  C  .  .  .  .  .  .
X.16     3 -1  /B  .   B  .   B   2  /B   2 -/A  C  -A /C  .  .  .  .  .  .
X.17     3 -1   2  .   2  .   B  /B  /B   B  -A /C -/A  C  .  .  .  .  .  .
X.18     3 -1   2  .   2  .  /B   B   B  /B -/A  C  -A /C  .  .  .  .  .  .
X.19     3 -1  /B  .   B  .   2   B   2  /B  -A /C -/A  C  .  .  .  .  .  .
X.20     3 -1   B  .  /B  .   2  /B   2   B -/A  C  -A /C  .  .  .  .  .  .

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