Properties

Label 18T137
Order \(324\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\wr S_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $137$
Group :  $C_3\wr S_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,4,10,17,7)(2,5,3,9,18,8)(11,13)(12,14)(15,16), (1,15,4,11,17,14)(2,16,3,12,18,13)(5,8,9)(6,7,10)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $((C_3^3:C_3):C_2):C_2$

Low degree siblings

9T24 x 3, 18T129 x 3, 18T136 x 3, 18T137 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $18$ $6$ $( 5,11, 8,14, 9,15)( 6,12, 7,13,10,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 2)( 3,17)( 4,18)( 5, 6)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 2)( 3,17)( 4,18)( 5,12)( 6,11)( 7,15)( 8,16)( 9,13)(10,14)$
$ 6, 6, 2, 2, 2 $ $54$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 8,16, 9,13)( 6,11, 7,15,10,14)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $18$ $6$ $( 1, 4,17)( 2, 3,18)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 4,17)( 2, 3,18)( 5,11, 8,14, 9,15)( 6,12, 7,13,10,16)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 4,17)( 2, 3,18)( 5,11, 9,15, 8,14)( 6,12,10,16, 7,13)$
$ 3, 3, 3, 3, 3, 3 $ $18$ $3$ $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$
$ 9, 9 $ $36$ $9$ $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$
$ 6, 6, 6 $ $54$ $6$ $( 1, 6,11, 2, 5,12)( 3, 9,13,17, 7,15)( 4,10,14,18, 8,16)$

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [324, 39]
Character table:   
      2  2  1  1  1  2  1  2  2  1  1  1  1  1  1  1  .  1
      3  4  3  3  3  2  2  1  1  1  4  3  2  2  2  2  2  1

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

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