↑

Properties

Label 18T178
Order \(576\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $178$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,12,2,13,11)(3,16,8,4,15,7)(5,17,10)(6,18,9), (1,9,16,2,10,15)(3,12,17,4,11,18)(5,7,14)(6,8,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$ x 3
36:  $C_3\times A_4$ x 3
144:  12T85 x 3

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

12T164 x 3, 18T178 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7, 8)( 9,10)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,12,13)( 2,11,14)( 3, 8,15)( 4, 7,16)( 5,10,17)( 6, 9,18)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,13,12)( 2,14,11)( 3,15, 8)( 4,16, 7)( 5,17,10)( 6,18, 9)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7, 8)( 9,10)(13,14)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,14)(15,16)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,12,13, 2,11,14)( 3, 8,15, 4, 7,16)( 5,10,17)( 6, 9,18)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,13,11, 2,14,12)( 3,15, 7, 4,16, 8)( 5,17,10)( 6,18, 9)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,16,10)( 2,15, 9)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,15,10)( 2,16, 9)( 3,18,12, 4,17,11)( 5,14, 8, 6,13, 7)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1, 7,17, 2, 8,18)( 3,10,13)( 4, 9,14)( 5,11,15, 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 8,18)( 2, 7,17)( 3, 9,14)( 4,10,13)( 5,11,15)( 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,10,16)( 2, 9,15)( 3,11,17)( 4,12,18)( 5, 7,14)( 6, 8,13)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1, 9,16)( 2,10,15)( 3,11,18, 4,12,17)( 5, 8,13, 6, 7,14)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,17, 8)( 2,18, 7)( 3,14,10, 4,13, 9)( 5,16,12, 6,15,11)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,18, 8)( 2,17, 7)( 3,14, 9)( 4,13,10)( 5,15,11)( 6,16,12)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 5, 4)( 2, 6, 3)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8664]
Character table: Data not available.