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Properties

Label 18T310
Order \(1296\)
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 $t$ :  $310$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,4,17,5,14,2,15,3,18,6,13)(7,10,11,8,9,12), (1,8,2,7)(3,12,4,11)(5,10,6,9)(13,15)(14,16)
$|\Aut(F/K)|$:  $2$

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
24:  $S_4$
36:  $S_3^2$
48:  $S_4\times C_2$
108:  $C_3^2 : D_{6} $
144:  12T83
324:  $((C_3^3:C_3):C_2):C_2$
432:  18T152

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $((C_3^3:C_3):C_2):C_2$

Low degree siblings

18T299 x 3, 18T310 x 2, 18T316 x 3, 18T317 x 3

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 $ $6$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(13,17,15)(14,18,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,11, 9)( 8,12,10)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 1, 5, 3)( 2, 6, 4)(13,17,15)(14,18,16)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)$
$ 6, 6, 3, 3 $ $6$ $6$ $( 1, 4, 5, 2, 3, 6)( 7,11, 9)( 8,12,10)(13,16,17,14,15,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
$ 6, 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 4, 5, 2, 3, 6)( 7,11, 9)( 8,12,10)(13,14)(15,16)(17,18)$
$ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $6$ $( 1, 2)( 3, 4)( 5, 6)(13,18,15,14,17,16)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 6, 3, 2, 5, 4)( 7, 9,11)( 8,10,12)(13,16,17,14,15,18)$
$ 6, 6, 3, 3 $ $6$ $6$ $( 1, 4, 5, 2, 3, 6)( 7, 9,11)( 8,10,12)(13,16,17,14,15,18)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $6$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9)( 8,12,10)(13,14)(15,16)(17,18)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $6$ $6$ $( 1, 6, 3, 2, 5, 4)(13,18,15,14,17,16)$
$ 6, 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 4, 5, 2, 3, 6)( 7, 9,11)( 8,10,12)(13,14)(15,16)(17,18)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $6$ $6$ $( 1, 6, 3, 2, 5, 4)(13,16,17,14,15,18)$
$ 3, 3, 3, 3, 3, 3 $ $72$ $3$ $( 1,10,16)( 2, 9,15)( 3,12,18)( 4,11,17)( 5, 8,14)( 6, 7,13)$
$ 9, 9 $ $144$ $9$ $( 1,12,16, 5,10,14, 3, 8,18)( 2,11,15, 6, 9,13, 4, 7,17)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 7,16,11,14, 9,18)( 8,15,12,13,10,17)$
$ 6, 6, 3, 3 $ $36$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,14,11,18, 9,16)( 8,13,12,17,10,15)$
$ 6, 6, 3, 3 $ $36$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,18,11,16, 9,14)( 8,17,12,15,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 7,18)( 8,17)( 9,14)(10,13)(11,16)(12,15)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $36$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,16)( 8,15)( 9,18)(10,17)(11,14)(12,13)$
$ 12, 6 $ $36$ $12$ $( 1, 4, 5, 2, 3, 6)( 7,17,12,16, 9,13, 8,18,11,15,10,14)$
$ 12, 2, 2, 2 $ $36$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7,15,12,14, 9,17, 8,16,11,13,10,18)$
$ 12, 6 $ $36$ $12$ $( 1, 6, 3, 2, 5, 4)( 7,13,12,18, 9,15, 8,14,11,17,10,16)$
$ 6, 4, 4, 4 $ $36$ $12$ $( 1, 4, 5, 2, 3, 6)( 7,13, 8,14)( 9,15,10,16)(11,17,12,18)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7,17, 8,18)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 3, 5)( 4, 6)( 7, 8)( 9,12)(10,11)(15,17)(16,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 4)( 2, 3)( 5, 6)( 7,12)( 8,11)( 9,10)(13,16)(14,15)(17,18)$
$ 6, 6, 6 $ $216$ $6$ $( 1,10,17, 2, 9,18)( 3, 8,13, 6,11,16)( 4, 7,14, 5,12,15)$
$ 4, 4, 4, 2, 2, 1, 1 $ $54$ $4$ $( 3, 5)( 4, 6)( 7,15, 8,16)( 9,13,10,14)(11,17,12,18)$
$ 12, 2, 2, 1, 1 $ $108$ $12$ $( 1, 5)( 2, 6)( 7,13,12,16, 9,17, 8,14,11,15,10,18)$
$ 6, 6, 2, 2, 2 $ $108$ $6$ $( 1, 4)( 2, 3)( 5, 6)( 7,18, 9,16,11,14)( 8,17,10,15,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7,16)( 8,15)( 9,14)(10,13)(11,18)(12,17)$

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1296, 1790]
Character table: Data not available.