Properties

Label 18T41
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 $t$ :  $41$
Group :  $C_2\times He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,17,13,4,16)(2,12,18,14,3,15)(5,9)(6,10)(7,8), (1,17)(2,18)(5,16,8,13,10,11)(6,15,7,14,9,12)
$|\Aut(F/K)|$:  $2$

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: $S_3$

Degree 6: $D_{6}$

Degree 9: $C_3^2 : C_6$

Low degree siblings

18T41, 18T42 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5, 8,10)( 6, 7, 9)(11,16,13)(12,15,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5,10, 8)( 6, 9, 7)(11,13,16)(12,14,15)$
$ 6, 6, 2, 2, 1, 1 $ $9$ $6$ $( 3,18)( 4,17)( 5,11, 8,16,10,13)( 6,12, 7,15, 9,14)$
$ 6, 6, 2, 2, 1, 1 $ $9$ $6$ $( 3,18)( 4,17)( 5,13,10,16, 8,11)( 6,14, 9,15, 7,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 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 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 7,10, 6, 8, 9)(11,15,13,12,16,14)(17,18)$
$ 6, 6, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 9, 8, 6,10, 7)(11,14,16,12,13,15)(17,18)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 8,15,10,14)( 6,11, 7,16, 9,13)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,14,10,15, 8,12)( 6,13, 9,16, 7,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 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 $ $6$ $6$ $( 1, 5,12, 2, 6,11)( 3, 7,13, 4, 8,14)( 9,16,17,10,15,18)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5,14, 2, 6,13)( 3, 7,16, 4, 8,15)( 9,11,17,10,12,18)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5,15, 2, 6,16)( 3, 7,11, 4, 8,12)( 9,13,17,10,14,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,12)( 2, 5,11)( 3, 8,13)( 4, 7,14)( 9,15,17)(10,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,14)( 2, 5,13)( 3, 8,16)( 4, 7,15)( 9,12,17)(10,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,15)( 2, 5,16)( 3, 8,11)( 4, 7,12)( 9,14,17)(10,13,18)$

Group invariants

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

        1a 3a 3b  6a  6b 2a 2b  6c  6d  6e  6f 2c 6g 3c  6h  6i 6j  3d  3e 3f
     2P 1a 3b 3a  3a  3b 1a 1a  3b  3a  3a  3b 1a 3c 3c  3e  3d 3f  3e  3d 3f
     3P 1a 1a 1a  2a  2a 2a 2b  2b  2b  2c  2c 2c 2b 1a  2b  2b 2b  1a  1a 1a
     5P 1a 3b 3a  6b  6a 2a 2b  6d  6c  6f  6e 2c 6g 3c  6i  6h 6j  3e  3d 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  -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  1
X.4      1  1  1   1   1  1 -1  -1  -1  -1  -1 -1 -1  1  -1  -1 -1   1   1  1
X.5      1  A /A -/A  -A -1 -1  -A -/A  /A   A  1 -1  1  -A -/A -1   A  /A  1
X.6      1 /A  A  -A -/A -1 -1 -/A  -A   A  /A  1 -1  1 -/A  -A -1  /A   A  1
X.7      1  A /A -/A  -A -1  1   A  /A -/A  -A -1  1  1   A  /A  1   A  /A  1
X.8      1 /A  A  -A -/A -1  1  /A   A  -A -/A -1  1  1  /A   A  1  /A   A  1
X.9      1  A /A  /A   A  1 -1  -A -/A -/A  -A -1 -1  1  -A -/A -1   A  /A  1
X.10     1 /A  A   A  /A  1 -1 -/A  -A  -A -/A -1 -1  1 -/A  -A -1  /A   A  1
X.11     1  A /A  /A   A  1  1   A  /A  /A   A  1  1  1   A  /A  1   A  /A  1
X.12     1 /A  A   A  /A  1  1  /A   A   A  /A  1  1  1  /A   A  1  /A   A  1
X.13     2  2  2   .   .  .  2   2   2   .   .  .  2  2  -1  -1 -1  -1  -1 -1
X.14     2  2  2   .   .  . -2  -2  -2   .   .  . -2  2   1   1  1  -1  -1 -1
X.15     2  B /B   .   .  .  2   B  /B   .   .  .  2  2 -/A  -A -1 -/A  -A -1
X.16     2 /B  B   .   .  .  2  /B   B   .   .  .  2  2  -A -/A -1  -A -/A -1
X.17     2  B /B   .   .  . -2  -B -/B   .   .  . -2  2  /A   A  1 -/A  -A -1
X.18     2 /B  B   .   .  . -2 -/B  -B   .   .  . -2  2   A  /A  1  -A -/A -1
X.19     6  .  .   .   .  . -6   .   .   .   .  .  3 -3   .   .  .   .   .  .
X.20     6  .  .   .   .  .  6   .   .   .   .  . -3 -3   .   .  .   .   .  .

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