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Properties

Label 18T362
Order \(2160\)
n \(18\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $362$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,6,12,16,13)(2,8,4,10,17,14)(3,9,5,11,18,15), (4,12)(5,10)(6,11)(7,15,16)(8,13,17)(9,14,18)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
720:  $S_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $S_6$

Degree 9: None

Low degree siblings

18T362

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 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $(13,17)(14,18)(15,16)$
$ 6, 3, 3, 3, 3 $ $15$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,16,14,17,15,18)$
$ 6, 3, 3, 3, 3 $ $15$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,18,15,17,14,16)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $(10,15,16)(11,13,17)(12,14,18)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,14,17)(11,15,18)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,13,18)(11,14,16)(12,15,17)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $45$ $2$ $( 7,10)( 8,11)( 9,12)(13,17)(14,18)(15,16)$
$ 6, 6, 3, 3 $ $45$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,12, 8,10, 9,11)(13,16,14,17,15,18)$
$ 6, 6, 3, 3 $ $45$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,11, 9,10, 8,12)(13,18,15,17,14,16)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $90$ $4$ $( 7,10,15,16)( 8,11,13,17)( 9,12,14,18)$
$ 12, 3, 3 $ $90$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7,12,13,16, 9,11,15,18, 8,10,14,17)$
$ 12, 3, 3 $ $90$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7,11,14,16, 8,12,15,17, 9,10,13,18)$
$ 3, 3, 3, 2, 2, 2, 1, 1, 1 $ $120$ $6$ $( 4, 9)( 5, 7)( 6, 8)(10,15,16)(11,13,17)(12,14,18)$
$ 6, 3, 3, 3, 3 $ $120$ $6$ $( 1, 3, 2)( 4, 8, 5, 9, 6, 7)(10,14,17)(11,15,18)(12,13,16)$
$ 6, 3, 3, 3, 3 $ $120$ $6$ $( 1, 2, 3)( 4, 7, 6, 9, 5, 8)(10,13,18)(11,14,16)(12,15,17)$
$ 5, 5, 5, 1, 1, 1 $ $144$ $5$ $( 4, 9,12,14,18)( 5, 7,10,15,16)( 6, 8,11,13,17)$
$ 15, 3 $ $144$ $15$ $( 1, 3, 2)( 4, 8,10,14,17, 5, 9,11,15,18, 6, 7,12,13,16)$
$ 15, 3 $ $144$ $15$ $( 1, 2, 3)( 4, 7,11,14,16, 6, 9,10,13,18, 5, 8,12,15,17)$
$ 6, 6, 6 $ $15$ $6$ $( 1, 6, 2, 4, 3, 5)( 7,12, 8,10, 9,11)(13,16,14,17,15,18)$
$ 6, 6, 6 $ $15$ $6$ $( 1, 5, 3, 4, 2, 6)( 7,11, 9,10, 8,12)(13,18,15,17,14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,17)(14,18)(15,16)$
$ 12, 6 $ $90$ $12$ $( 1, 6, 2, 4, 3, 5)( 7,12,13,16, 9,11,15,18, 8,10,14,17)$
$ 12, 6 $ $90$ $12$ $( 1, 5, 3, 4, 2, 6)( 7,11,14,16, 8,12,15,17, 9,10,13,18)$
$ 4, 4, 4, 2, 2, 2 $ $90$ $4$ $( 1, 4)( 2, 5)( 3, 6)( 7,10,15,16)( 8,11,13,17)( 9,12,14,18)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 6, 7)( 2, 4, 8)( 3, 5, 9)(10,14,17)(11,15,18)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5, 8)( 2, 6, 9)( 3, 4, 7)(10,13,18)(11,14,16)(12,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 4, 9)( 2, 5, 7)( 3, 6, 8)(10,15,16)(11,13,17)(12,14,18)$
$ 6, 6, 6 $ $120$ $6$ $( 1, 6, 7,12,13,16)( 2, 4, 8,10,14,17)( 3, 5, 9,11,15,18)$
$ 6, 6, 6 $ $120$ $6$ $( 1, 5, 8,12,15,17)( 2, 6, 9,10,13,18)( 3, 4, 7,11,14,16)$
$ 6, 6, 6 $ $120$ $6$ $( 1, 4, 9,12,14,18)( 2, 5, 7,10,15,16)( 3, 6, 8,11,13,17)$

Group invariants

Order:  $2160=2^{4} \cdot 3^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.