Properties

Label 18T46
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $46$
Group :  $C_3\times S_3^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,13,3,7,15)(2,8,14)(4,12,16,6,10,18)(5,11,17), (1,11,9,16,14,6)(2,12,7,17,15,4)(3,10,8,18,13,5), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,18,14,16,15,17)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$ x 2, $C_6$ x 3
12:  $D_{6}$ x 2, $C_6\times C_2$
18:  $S_3\times C_3$ x 2
36:  $S_3^2$, $C_6\times S_3$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$, $S_3\times C_3$

Degree 9: None

Low degree siblings

12T70, 18T43, 18T46

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 38]
Character table: Data not available.