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Properties

Label 18T16
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_9\times S_3$

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Group action invariants

Degree $n$:  $18$
Transitive number $t$:  $16$
Group:  $C_9\times S_3$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $9$
Generators:  (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,12)(8,10)(9,11), (1,18,9,13,4,11,2,16,7,14,5,12,3,17,8,15,6,10)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$9$:  $C_9$
$18$:  $S_3\times C_3$, $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: None

Low degree siblings

27T12

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  no
Abelian:  no
Solvable:  yes
Label:  54.4
Character table: not available.