Properties

Label 18T2
Order \(18\)
n \(18\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_6 \times C_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $2$
Group :  $C_6 \times C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,7,13,2,8,14)(3,9,15,4,10,16)(5,11,17,6,12,18), (1,16,5,2,15,6)(3,11,8,4,12,7)(9,17,14,10,18,13)
$|\Aut(F/K)|$:  $18$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$ x 4

Degree 6: $C_6$ x 4

Degree 9: $C_3^2$

Low degree siblings

There are no siblings 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, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,14,16)(12,13,15)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 4,17, 2, 3,18)( 5, 7,10, 6, 8, 9)(11,13,16,12,14,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 5,15)( 2, 6,16)( 3, 8,12)( 4, 7,11)( 9,14,18)(10,13,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 6,15, 2, 5,16)( 3, 7,12, 4, 8,11)( 9,13,18,10,14,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 7,13, 2, 8,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 9,12, 2,10,11)( 3, 6,13, 4, 5,14)( 7,15,18, 8,16,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,10,12)( 2, 9,11)( 3, 5,13)( 4, 6,14)( 7,16,18)( 8,15,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1,11,10, 2,12, 9)( 3,14, 5, 4,13, 6)( 7,17,16, 8,18,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,12,10)( 2,11, 9)( 3,13, 5)( 4,14, 6)( 7,18,16)( 8,17,15)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,13, 8)( 2,14, 7)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$
$ 6, 6, 6 $ $1$ $6$ $( 1,14, 8, 2,13, 7)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,15, 5)( 2,16, 6)( 3,12, 8)( 4,11, 7)( 9,18,14)(10,17,13)$
$ 6, 6, 6 $ $1$ $6$ $( 1,16, 5, 2,15, 6)( 3,11, 8, 4,12, 7)( 9,17,14,10,18,13)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 3)( 2,18, 4)( 5,10, 8)( 6, 9, 7)(11,16,14)(12,15,13)$
$ 6, 6, 6 $ $1$ $6$ $( 1,18, 3, 2,17, 4)( 5, 9, 8, 6,10, 7)(11,15,14,12,16,13)$

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [18, 5]
Character table:   
      2  1  1  1   1  1   1   1  1   1  1   1  1  1   1  1   1  1   1
      3  2  2  2   2  2   2   2  2   2  2   2  2  2   2  2   2  2   2

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

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

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