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Properties

Label 18T201
Order \(648\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $201$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,10,18,15,6,3,13,8,2,12,9,17,16,5,4,14,7), (1,13,8,4,12,9,17,16,5,2,14,7,3,11,10,18,15,6)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
324:  $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

Low degree siblings

18T199 x 2, 18T205

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

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 706]
Character table: Data not available.