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Properties

Label 18T42
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times He_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $42$
Group :  $C_2\times He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,4)(5,7,10,6,8,9)(11,15,13,12,16,14)(17,18), (1,5,16,2,6,15)(3,9,12,17,8,13)(4,10,11,18,7,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
36:  $C_6\times S_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_3^2 : S_3 $

Low degree siblings

18T41 x 2, 18T42

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 25]
Character table:    
      2  2  1  2  2  1  2  1  1   1   2   2   1   2   2   1   2   2   1   2
      3  3  2  1  3  2  1  3  3   2   2   1   2   2   1   2   2   1   2   2

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

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

      2   2
      3   1

         6j
     2P  3d
     3P  2c
     5P  6e

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

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