Properties

Label 18T14
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_9:C_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $14$
Group :  $C_2\times C_9:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,17,15,13,11,9,8,6,3,2,18,16,14,12,10,7,5,4), (1,3,11,8,10,18,14,15,5)(2,4,12,7,9,17,13,16,6)
$|\Aut(F/K)|$:  $6$

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:  $C_3^2$
18:  $C_6 \times C_3$
27:  $C_9:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_9:C_3$

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 11]
Character table: Data not available.