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Properties

Label 18T520
Order \(6912\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $520$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,6,17,4,15)(2,14,5,18,3,16)(7,10,12)(8,9,11), (1,10,3,8,6,11)(2,9,4,7,5,12)(13,16)(14,15)
$|\Aut(F/K)|$:  $2$

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: None

Degree 3: $S_3$

Degree 6: None

Degree 9: $C_3^2 : D_{6} $

Low degree siblings

12T268 x 2, 18T517 x 2, 18T518 x 2, 18T519 x 2, 18T520

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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,12,10)( 8,11, 9)(13,17,15)(14,18,16)$
$ 3, 3, 3, 3, 3, 3 $ $192$ $3$ $( 1,18, 9)( 2,17,10)( 3,13,11)( 4,14,12)( 5,16, 8)( 6,15, 7)$
$ 6, 6, 3, 3 $ $576$ $6$ $( 1,18,10, 2,17, 9)( 3,14,12, 4,13,11)( 5,15, 8)( 6,16, 7)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $288$ $6$ $( 1, 5, 3)( 2, 6, 4)( 9,10)(11,12)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $96$ $3$ $( 1, 5, 3)( 2, 6, 4)(13,15,17)(14,16,18)$
$ 6, 6, 3, 3 $ $288$ $6$ $( 1,16, 9, 2,15,10)( 3,18,12)( 4,17,11)( 5,13, 7, 6,14, 8)$
$ 3, 3, 3, 3, 3, 3 $ $96$ $3$ $( 1,16, 9)( 2,15,10)( 3,17,11)( 4,18,12)( 5,14, 8)( 6,13, 7)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $36$ $2$ $( 1,17)( 2,18)( 3,14)( 4,13)( 5,16)( 6,15)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $216$ $4$ $( 1,17)( 2,18)( 3,13, 4,14)( 5,15, 6,16)( 7, 8)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $108$ $2$ $( 1,17)( 2,18)( 3,14)( 4,13)( 5,16)( 6,15)( 9,10)(11,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ $108$ $4$ $( 1,18, 2,17)( 3,13, 4,14)( 5,16)( 6,15)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $108$ $4$ $( 1,18, 2,17)( 3,14)( 4,13)( 5,15, 6,16)( 7, 8)(11,12)$
$ 6, 6, 3, 3 $ $1152$ $6$ $( 1,15, 4,17, 6,13)( 2,16, 3,18, 5,14)( 7,12,10)( 8,11, 9)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $216$ $4$ $( 1, 6)( 2, 5)( 9,11)(10,12)(13,18,14,17)(15,16)$
$ 4, 4, 4, 2, 2, 2 $ $72$ $4$ $( 1, 5, 2, 6)( 3, 4)( 7, 8)( 9,11,10,12)(13,18,14,17)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $72$ $2$ $( 1, 5)( 2, 6)( 9,11)(10,12)(13,17)(14,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $216$ $4$ $( 1, 5)( 2, 6)( 7, 8)( 9,11,10,12)(13,17,14,18)(15,16)$
$ 6, 6, 3, 3 $ $576$ $6$ $( 1,15, 8, 5,17,11)( 2,16, 7, 6,18,12)( 3,13, 9)( 4,14,10)$
$ 12, 6 $ $576$ $12$ $( 1,15, 8, 5,17,12, 2,16, 7, 6,18,11)( 3,13,10, 4,14, 9)$
$ 6, 6, 2, 2, 1, 1 $ $576$ $6$ $( 1,15, 5,18, 3,14)( 2,16, 6,17, 4,13)( 9,11)(10,12)$
$ 6, 6, 4, 2 $ $576$ $12$ $( 1,16, 6,17, 4,14)( 2,15, 5,18, 3,13)( 7, 8)( 9,11,10,12)$
$ 4, 4, 4, 2, 2, 2 $ $216$ $4$ $( 1, 6, 2, 5)( 3, 4)( 7,15, 8,16)( 9,13,10,14)(11,18)(12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $72$ $2$ $( 1, 5)( 2, 6)( 7,16)( 8,15)( 9,14)(10,13)(11,18)(12,17)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $216$ $4$ $( 1, 5)( 2, 6)( 7,16)( 8,15)( 9,14,10,13)(11,18,12,17)$
$ 4, 2, 2, 2, 2, 2, 2, 2 $ $72$ $4$ $( 1, 5, 2, 6)( 3, 4)( 7,16)( 8,15)( 9,13)(10,14)(11,17)(12,18)$

Group invariants

Order:  $6912=2^{8} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.