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Properties

Label 18T268
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 $t$ :  $268$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5)(2,4,6)(7,17,12,16,10,13)(8,18,11,15,9,14), (1,18,4,14,5,16)(2,17,3,13,6,15)(7,12,9)(8,11,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$
72:  12T43
288:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: None

Degree 9: $S_3\times C_3$

Low degree siblings

12T206 x 3, 18T268 x 2, 18T271 x 3

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 $ $9$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)$
$ 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 $ $6$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,14)(15,16)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,17,16)(14,18,15)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,16,17)(14,15,18)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,13,11)( 2,14,12)( 3,15, 8)( 4,16, 7)( 5,17, 9)( 6,18,10)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,14,11, 2,13,12)( 3,15, 7, 4,16, 8)( 5,18,10)( 6,17, 9)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,17, 8)( 2,18, 7)( 3,14,10, 4,13, 9)( 5,16,12, 6,15,11)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,17, 7)( 2,18, 8)( 3,13,10)( 4,14, 9)( 5,15,11)( 6,16,12)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,16,10, 2,15, 9)( 3,18,12, 4,17,11)( 5,13, 8)( 6,14, 7)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,15,10)( 2,16, 9)( 3,18,12)( 4,17,11)( 5,14, 8)( 6,13, 7)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 7,16)( 8,15)( 9,18)(10,17)(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 1, 2)( 5, 6)( 7,16, 8,15)( 9,17,10,18)(11,13)(12,14)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ $36$ $4$ $( 7,15, 8,16)( 9,18,10,17)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $36$ $2$ $( 1, 2)( 5, 6)( 7,15)( 8,16)( 9,17)(10,18)(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 3, 4)( 5, 6)( 7,16)( 8,15)( 9,17,10,18)(11,13,12,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 1, 2)( 3, 4)( 7,16, 8,15)( 9,18)(10,17)(11,14,12,13)$
$ 6, 6, 3, 3 $ $192$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,13,10,16,12,17)( 8,14, 9,15,11,18)$
$ 6, 6, 3, 3 $ $192$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,17,12,16,10,13)( 8,18,11,15, 9,14)$

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.