Properties

Label 9T15
Order \(72\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_3^2:C_8$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $15$
Group :  $C_3^2:C_8$
CHM label :  $E(9):8$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,4,5,2,3,8,7), (1,2,9)(3,4,5)(6,7,8), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T46, 18T28, 24T81, 36T49

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 $ $1$ $1$ $()$
$ 8, 1 $ $9$ $8$ $(2,3,6,7,9,8,5,4)$
$ 8, 1 $ $9$ $8$ $(2,4,5,8,9,7,6,3)$
$ 4, 4, 1 $ $9$ $4$ $(2,5,9,6)(3,4,8,7)$
$ 4, 4, 1 $ $9$ $4$ $(2,6,9,5)(3,7,8,4)$
$ 8, 1 $ $9$ $8$ $(2,7,5,3,9,4,6,8)$
$ 8, 1 $ $9$ $8$ $(2,8,6,4,9,3,5,7)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 39]
Character table:   
     2  3   3   3  3  3   3   3  3  .
     3  2   .   .  .  .   .   .  .  2

       1a  8a  8b 4a 4b  8c  8d 2a 3a
    2P 1a  4b  4a 2a 2a  4a  4b 1a 3a
    3P 1a  8c  8d 4b 4a  8a  8b 2a 1a
    5P 1a  8d  8c 4a 4b  8b  8a 2a 3a
    7P 1a  8b  8a 4b 4a  8d  8c 2a 3a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)