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Properties

Label 18T109
Order \(288\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times A_4^2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $109$
Group :  $C_2\times A_4^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,18)(2,9,17)(3,12,14)(4,11,13)(5,7,16)(6,8,15), (1,6,3)(2,5,4)(7,12,10,8,11,9)(13,18,15)(14,17,16)
$|\Aut(F/K)|$:  $2$

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:  $A_4$ x 2
18:  $C_6 \times C_3$
24:  $A_4\times C_2$ x 2
36:  $C_3\times A_4$ x 2
72:  18T25 x 2
144:  12T85

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: None

Degree 9: $C_3^2$

Low degree siblings

There are no siblings 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 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 5, 6)(11,12)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)$
$ 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, 3, 3 $ $12$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12, 8,10,11)(13,15,17)(14,16,18)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12, 8,10,11)(13,16,17,14,15,18)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 3, 5, 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,17,14,16,18)$
$ 6, 3, 3, 3, 3 $ $12$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10, 8,12, 9)(13,17,15)(14,18,16)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10, 8,12, 9)(13,18,15,14,17,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,17,15)(14,18,16)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,12, 9, 8,11,10)(13,17,16,14,18,15)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 7,13)( 2, 8,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$
$ 6, 3, 3, 3, 3 $ $12$ $6$ $( 1, 7,13, 2, 8,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 7,13)( 2, 8,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 7,13, 2, 8,14)( 3,10,15, 4, 9,16)( 5,12,17, 6,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 9,18)( 2,10,17)( 3,11,13)( 4,12,14)( 5, 7,15)( 6, 8,16)$
$ 6, 6, 6 $ $16$ $6$ $( 1, 9,18, 2,10,17)( 3,11,13, 4,12,14)( 5, 7,15, 6, 8,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,11,16)( 2,12,15)( 3, 7,17)( 4, 8,18)( 5, 9,13)( 6,10,14)$
$ 6, 6, 6 $ $16$ $6$ $( 1,11,16, 2,12,15)( 3, 7,17, 4, 8,18)( 5, 9,13, 6,10,14)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,13, 7)( 2,14, 8)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$
$ 6, 6, 6 $ $4$ $6$ $( 1,13, 8, 2,14, 7)( 3,15, 9, 4,16,10)( 5,17,11, 6,18,12)$
$ 6, 3, 3, 3, 3 $ $12$ $6$ $( 1,13, 7)( 2,14, 8)( 3,15,10)( 4,16, 9)( 5,18,12, 6,17,11)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1,13, 8, 2,14, 7)( 3,15, 9, 4,16,10)( 5,18,11)( 6,17,12)$
$ 6, 6, 6 $ $16$ $6$ $( 1,15,11, 2,16,12)( 3,17, 8, 4,18, 7)( 5,13, 9, 6,14,10)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,15,12)( 2,16,11)( 3,17, 7)( 4,18, 8)( 5,13,10)( 6,14, 9)$
$ 6, 6, 6 $ $16$ $6$ $( 1,17, 9, 2,18,10)( 3,13,11, 4,14,12)( 5,15, 8, 6,16, 7)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,17,10)( 2,18, 9)( 3,13,12)( 4,14,11)( 5,15, 7)( 6,16, 8)$

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [288, 1029]
Character table: Data not available.