Properties

Label 18T5
Order \(18\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $5$
Group :  $D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,6,8,10,12,14,16,17)(2,3,5,7,9,11,13,15,18), (1,15)(2,16)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)
$|\Aut(F/K)|$:  $18$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $D_{9}$

Low degree siblings

9T3

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

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [18, 1]
Character table:   
     2  1  1  .  .  .  .
     3  2  .  2  2  2  2

       1a 2a 9a 9b 3a 9c
    2P 1a 1a 9b 9c 3a 9a
    3P 1a 2a 3a 3a 1a 3a
    5P 1a 2a 9c 9a 3a 9b
    7P 1a 2a 9b 9c 3a 9a

X.1     1  1  1  1  1  1
X.2     1 -1  1  1  1  1
X.3     2  . -1 -1  2 -1
X.4     2  .  A  B -1  C
X.5     2  .  B  C -1  A
X.6     2  .  C  A -1  B

A = E(9)^2+E(9)^7
B = E(9)^4+E(9)^5
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7