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Properties

Label 18T92
Order \(216\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3^2.A_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $92$
Group :  $C_2\times C_3^2.A_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,18,5,11,16,3,9,14,2,7,17,6,12,15,4,10,13), (1,7,14,5,12,17,3,10,15,2,8,13,6,11,18,4,9,16)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
12:  $A_4$
18:  $C_6 \times C_3$
24:  $A_4\times C_2$
27:  $C_9:C_3$
36:  $C_3\times A_4$
54:  18T14
72:  18T25
108:  18T47

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 9: $C_9:C_3$

Low degree siblings

18T92 x 2

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

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 106]
Character table: Data not available.