Properties

Label 18T39
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2:D_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $39$
Group :  $C_2^2:D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,13,5,8,18,3,11,16)(2,9,14,6,7,17,4,12,15), (1,18)(2,17)(3,16)(4,15)(5,13)(6,14)(7,10)(8,9)(11,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
18:  $D_{9}$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 9: $D_{9}$

Low degree siblings

18T38

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 15]
Character table:   
     2  3  3  2  2  2  2  .  .  .
     3  2  1  .  .  2  1  2  2  2

       1a 2a 4a 2b 3a 6a 9a 9b 9c
    2P 1a 1a 2a 1a 3a 3a 9b 9c 9a
    3P 1a 2a 4a 2b 1a 2a 3a 3a 3a
    5P 1a 2a 4a 2b 3a 6a 9c 9a 9b
    7P 1a 2a 4a 2b 3a 6a 9b 9c 9a

X.1     1  1  1  1  1  1  1  1  1
X.2     1  1 -1 -1  1  1  1  1  1
X.3     2  2  .  .  2  2 -1 -1 -1
X.4     2  2  .  . -1 -1  A  C  B
X.5     2  2  .  . -1 -1  B  A  C
X.6     2  2  .  . -1 -1  C  B  A
X.7     3 -1 -1  1  3 -1  .  .  .
X.8     3 -1  1 -1  3 -1  .  .  .
X.9     6 -2  .  . -3  1  .  .  .

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