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Properties

Label 18T86
Order \(162\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\wr S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $86$
Group :  $C_3\wr S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,15,17,8,13,3,6,11)(2,9,16,18,7,14,4,5,12), (1,16,17,14,3,12)(2,15,18,13,4,11)(5,8,9,6,7,10)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $C_3 \wr S_3 $

Low degree siblings

9T20 x 3, 18T86 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 $ $3$ $3$ $(11,13,15)(12,14,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $(11,15,13)(12,16,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 7, 9)( 6, 8,10)(11,15,13)(12,16,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)(17,18)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3, 4)( 5,11, 7,13, 9,15)( 6,12, 8,14,10,16)(17,18)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3, 4)( 5,11, 9,15, 7,13)( 6,12,10,16, 8,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
$ 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 7, 9)( 6, 8,10)(11,15,13)(12,16,14)$
$ 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$
$ 6, 2, 2, 2, 2, 2, 2 $ $9$ $6$ $( 1, 4,17, 2, 3,18)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 4,17, 2, 3,18)( 5,11, 7,13, 9,15)( 6,12, 8,14,10,16)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 4,17, 2, 3,18)( 5,11, 9,15, 7,13)( 6,12,10,16, 8,14)$
$ 6, 2, 2, 2, 2, 2, 2 $ $9$ $6$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,17)(10,18)(11,16,13,12,15,14)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 5, 3, 7,17, 9)( 2, 6, 4, 8,18,10)(11,16,13,12,15,14)$
$ 6, 6, 6 $ $9$ $6$ $( 1, 5,17, 9, 3, 7)( 2, 6,18,10, 4, 8)(11,16,13,12,15,14)$
$ 3, 3, 3, 3, 3, 3 $ $18$ $3$ $( 1, 6,11)( 2, 5,12)( 3, 8,13)( 4, 7,14)( 9,16,18)(10,15,17)$
$ 9, 9 $ $18$ $9$ $( 1, 6,11, 3, 8,13,17,10,15)( 2, 5,12, 4, 7,14,18, 9,16)$
$ 9, 9 $ $18$ $9$ $( 1, 6,11,17,10,15, 3, 8,13)( 2, 5,12,18, 9,16, 4, 7,14)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 10]
Character table: Data not available.