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Properties

Label 18T150
Order \(432\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times S_3\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $150$
Group :  $S_3\times S_3\wr C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(5,6)(8,9)(11,12)(14,15)(17,18), (1,6,2,4,3,5)(7,12,14,16,9,11,13,18,8,10,15,17), (1,3,2)(4,18,11)(5,16,12)(6,17,10)(7,15,8,13,9,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 3
16:  $D_4\times C_2$
24:  $S_3 \times C_2^2$
48:  12T28
72:  $C_3^2:D_4$
144:  12T77

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$, $C_3^2:D_4$

Degree 9: None

Low degree siblings

12T156 x 2, 18T150

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(10,16)(11,17)(12,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4,10,16)( 5,11,17)( 6,12,18)$
$ 3, 3, 3, 2, 2, 2, 1, 1, 1 $ $12$ $6$ $( 4,10,16)( 5,11,17)( 6,12,18)( 7,13)( 8,14)( 9,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $18$ $2$ $( 2, 3)( 5, 6)( 8, 9)(10,16)(11,18)(12,17)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $27$ $2$ $( 2, 3)( 5, 6)( 7,13)( 8,15)( 9,14)(10,16)(11,18)(12,17)$
$ 6, 3, 2, 2, 2, 1, 1, 1 $ $12$ $6$ $( 2, 3)( 4,10,16)( 5,12,17, 6,11,18)( 8, 9)(14,15)$
$ 6, 3, 2, 2, 2, 2, 1 $ $36$ $6$ $( 2, 3)( 4,10,16)( 5,12,17, 6,11,18)( 7,13)( 8,15)( 9,14)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 6, 3, 3, 3, 3 $ $12$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,17,12,16,11,18)(13,14,15)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,14, 9,13, 8,15)(10,17,12,16,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4,11,18)( 5,12,16)( 6,10,17)( 7, 8, 9)(13,14,15)$
$ 6, 3, 3, 3, 3 $ $24$ $6$ $( 1, 2, 3)( 4,11,18)( 5,12,16)( 6,10,17)( 7,14, 9,13, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,16)(14,17)(15,18)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 4)( 2, 5)( 3, 6)( 7,10,13,16)( 8,11,14,17)( 9,12,15,18)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 4, 7,10,13,16)( 2, 5, 8,11,14,17)( 3, 6, 9,12,15,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 4)( 2, 6)( 3, 5)( 7,10)( 8,12)( 9,11)(13,16)(14,18)(15,17)$
$ 4, 4, 4, 2, 2, 2 $ $54$ $4$ $( 1, 4)( 2, 6)( 3, 5)( 7,10,13,16)( 8,12,14,18)( 9,11,15,17)$
$ 6, 6, 6 $ $36$ $6$ $( 1, 4, 7,10,13,16)( 2, 6, 8,12,14,18)( 3, 5, 9,11,15,17)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 5, 3, 4, 2, 6)( 7,11, 9,10, 8,12)(13,17,15,16,14,18)$
$ 12, 6 $ $36$ $12$ $( 1, 5, 3, 4, 2, 6)( 7,11,15,16, 8,12,13,17, 9,10,14,18)$
$ 6, 6, 6 $ $24$ $6$ $( 1, 5, 9,10,14,18)( 2, 6, 7,11,15,16)( 3, 4, 8,12,13,17)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 7,13)( 2, 9,14, 3, 8,15)( 4,10,16)( 5,12,17, 6,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 8,15)( 2, 9,13)( 3, 7,14)( 4,11,18)( 5,12,16)( 6,10,17)$

Group invariants

Order:  $432=2^{4} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [432, 741]
Character table: Data not available.