↑

Properties

Label 18T646
Order \(20736\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $646$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,9,2,14,10)(3,16,11)(4,15,12)(5,17,8,6,18,7), (1,18,12,4,15,9,6,13,7)(2,17,11,3,16,10,5,14,8)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

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

Low degree siblings

12T284, 18T645, 18T647 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 1, 2)( 3, 4)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(15,16)(17,18)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $3$ $( 1, 5, 3)( 2, 6, 4)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $72$ $6$ $( 1, 5, 3)( 2, 6, 4)(15,16)(17,18)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $72$ $6$ $( 1, 5, 3)( 2, 6, 4)( 9,10)(11,12)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $216$ $6$ $( 1, 5, 3)( 2, 6, 4)( 9,10)(11,12)(15,16)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $192$ $3$ $( 1, 5, 3)( 2, 6, 4)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $576$ $6$ $( 1, 5, 3)( 2, 6, 4)( 9,10)(11,12)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $256$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $256$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $108$ $2$ $( 3, 6)( 4, 5)( 7,12)( 8,11)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $108$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,12)( 8,11)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $324$ $2$ $( 3, 6)( 4, 5)( 7,12)( 8,11)(15,16)(17,18)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $324$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,12)( 8,11)(15,16)(17,18)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $108$ $4$ $( 3, 6)( 4, 5)( 7,11, 8,12)( 9,10)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ $108$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,11, 8,12)( 9,10)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $324$ $4$ $( 3, 6)( 4, 5)( 7,11, 8,12)( 9,10)(15,16)(17,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $324$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,11, 8,12)( 9,10)(15,16)(17,18)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $864$ $6$ $( 3, 6)( 4, 5)( 7,12)( 8,11)(13,15,17)(14,16,18)$
$ 4, 3, 3, 2, 2, 2, 1, 1 $ $864$ $12$ $( 1, 2)( 3, 6, 4, 5)( 7,12)( 8,11)(13,15,17)(14,16,18)$
$ 4, 3, 3, 2, 2, 2, 1, 1 $ $864$ $12$ $( 3, 6)( 4, 5)( 7,11, 8,12)( 9,10)(13,15,17)(14,16,18)$
$ 4, 4, 3, 3, 2, 2 $ $864$ $12$ $( 1, 2)( 3, 6, 4, 5)( 7,11, 8,12)( 9,10)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $576$ $3$ $( 1, 9,14)( 2,10,13)( 3,11,16)( 4,12,15)( 5, 8,18)( 6, 7,17)$
$ 6, 6, 3, 3 $ $1728$ $6$ $( 1, 9,14, 2,10,13)( 3,11,16, 4,12,15)( 5, 8,18)( 6, 7,17)$
$ 9, 9 $ $2304$ $9$ $( 1, 9,14, 5, 8,18, 3,11,16)( 2,10,13, 6, 7,17, 4,12,15)$
$ 9, 9 $ $2304$ $9$ $( 1, 9,14, 3,11,16, 5, 8,18)( 2,10,13, 4,12,15, 6, 7,17)$
$ 3, 3, 3, 3, 3, 3 $ $576$ $3$ $( 1,14, 9)( 2,13,10)( 3,16,11)( 4,15,12)( 5,18, 8)( 6,17, 7)$
$ 6, 6, 3, 3 $ $1728$ $6$ $( 1,14, 9, 2,13,10)( 3,16,11, 4,15,12)( 5,18, 8)( 6,17, 7)$
$ 9, 9 $ $2304$ $9$ $( 1,14, 9, 5,18, 8, 3,16,11)( 2,13,10, 6,17, 7, 4,15,12)$
$ 9, 9 $ $2304$ $9$ $( 1,14, 9, 3,16,11, 5,18, 8)( 2,13,10, 4,15,12, 6,17, 7)$

Group invariants

Order:  $20736=2^{8} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.