Properties

Label 18T48
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6^2:C_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $48$
Group :  $C_6^2:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,18)(2,10,17)(3,12,13)(4,11,14)(5,7,16)(6,8,15), (1,13,10)(2,14,9)(3,16,11)(4,15,12)(5,18,8)(6,17,7)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

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

Low degree siblings

18T48 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 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 7, 9,12)( 8,10,11)(13,18,16)(14,17,15)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $3$ $6$ $( 7,10,12, 8, 9,11)(13,17,16,14,18,15)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $3$ $6$ $( 7,11, 9, 8,12,10)(13,15,18,14,16,17)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 7,12, 9)( 8,11,10)(13,16,18)(14,15,17)$
$ 6, 3, 3, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,12)( 8,10,11)(13,17,16,14,18,15)$
$ 6, 3, 3, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,12, 8, 9,11)(13,18,16)(14,17,15)$
$ 6, 3, 3, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9, 8,12,10)(13,16,18)(14,15,17)$
$ 6, 3, 3, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,12, 9)( 8,11,10)(13,15,18,14,16,17)$
$ 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 $ $12$ $3$ $( 1, 7,13)( 2, 8,14)( 3, 9,16)( 4,10,15)( 5,12,18)( 6,11,17)$
$ 3, 3, 3, 3, 3, 3 $ $12$ $3$ $( 1, 7,15)( 2, 8,16)( 3, 9,17)( 4,10,18)( 5,12,14)( 6,11,13)$
$ 3, 3, 3, 3, 3, 3 $ $12$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,15)( 6,11,16)$
$ 3, 3, 3, 3, 3, 3 $ $12$ $3$ $( 1,13, 7)( 2,14, 8)( 3,16, 9)( 4,15,10)( 5,18,12)( 6,17,11)$
$ 3, 3, 3, 3, 3, 3 $ $12$ $3$ $( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,18, 9)( 6,17,10)$
$ 3, 3, 3, 3, 3, 3 $ $12$ $3$ $( 1,13, 9)( 2,14,10)( 3,16,12)( 4,15,11)( 5,18, 7)( 6,17, 8)$

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 22]
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  3   2   2  3  2  2  2  2  2  2

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

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

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