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Properties

Label 18T314
Order \(1296\)
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 $t$ :  $314$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,9,3,15,6)(2,11,10,4,16,5)(7,17,13,8,18,14), (1,16,17,12,4,13)(2,15,18,11,3,14)(5,8,9)(6,7,10)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $S_3\wr S_3$

Low degree siblings

9T31, 12T213, 18T300, 18T303, 18T311, 18T312, 18T315, 18T319, 18T320

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 5, 8, 9)( 6, 7,10)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 1, 3)( 2, 4)(11,12)(13,15)(14,16)(17,18)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $54$ $6$ $( 1, 3)( 2, 4)( 5, 8, 9)( 6, 7,10)(11,12)(13,15)(14,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $72$ $3$ $( 1, 9,15)( 2,10,16)( 3, 6,12)( 4, 5,11)( 7,13,18)( 8,14,17)$
$ 9, 9 $ $144$ $9$ $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(11,12)(13,15)(14,16)$
$ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 5, 8, 9)( 6, 7,10)(11,12)(13,15)(14,16)$
$ 3, 3, 3, 3, 2, 2, 2 $ $36$ $6$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)(11,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 2)( 3,17)( 4,18)( 5, 7)( 6, 8)( 9,10)(11,13)(12,14)(15,16)$
$ 6, 6, 6 $ $216$ $6$ $( 1, 9,15,18, 7,13)( 2,10,16,17, 8,14)( 3, 6,12, 4, 5,11)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 1,16,17,12, 4,13)( 2,15,18,11, 3,14)$
$ 6, 6, 3, 3 $ $72$ $6$ $( 1,16,17,12, 4,13)( 2,15,18,11, 3,14)( 5, 8, 9)( 6, 7,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1,16)( 2,15)( 3,14)( 4,13)(11,18)(12,17)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $36$ $6$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 8, 9)( 6, 7,10)(11,18)(12,17)$
$ 4, 4, 4, 2, 2, 2 $ $162$ $4$ $( 1,11,18,16)( 2,12,17,15)( 3,13, 4,14)( 5, 7)( 6, 8)( 9,10)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $54$ $4$ $( 1,14, 2,13)( 3,16,17,11)( 4,15,18,12)$
$ 4, 4, 4, 3, 3 $ $108$ $12$ $( 1,14, 2,13)( 3,16,17,11)( 4,15,18,12)( 5, 8, 9)( 6, 7,10)$
$ 6, 6, 2, 2, 2 $ $108$ $6$ $( 1,13, 4,12,17,16)( 2,14, 3,11,18,15)( 5, 7)( 6, 8)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5, 7)( 6, 8)( 9,10)(15,18)(16,17)$

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1296, 3490]
Character table: Data not available.