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Properties

Label 18T264
Order \(1152\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $264$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,8,5,13,12,3,17,10,2,16,7,6,14,11,4,18,9), (1,6,4)(2,5,3)(7,11,10)(8,12,9)(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:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$
576:  12T166

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $C_9$

Low degree siblings

18T264 x 6

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

Group invariants

Order:  $1152=2^{7} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.