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Properties

Label 9T16
Order \(72\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $S_3^2:C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $16$
Group :  $S_3^2:C_2$
CHM label :  $E(9):D_{8}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5)(6,7), (1,8)(2,4)(5,7), (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:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

6T13 x 2, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1 $ $6$ $2$ $(3,7)(4,8)(5,6)$
$ 2, 2, 2, 1, 1, 1 $ $6$ $2$ $(2,5)(4,7)(6,9)$
$ 4, 4, 1 $ $18$ $4$ $(2,5,9,6)(3,4,8,7)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 6, 3 $ $12$ $6$ $(1,2,3,4,8,6)(5,9,7)$
$ 3, 3, 3 $ $4$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 6, 3 $ $12$ $6$ $(1,2,9)(3,8,5,7,4,6)$
$ 3, 3, 3 $ $4$ $3$ $(1,3,8)(2,4,6)(5,7,9)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 40]
Character table:    
     2  3  2  2  2  3  1  1  1  1
     3  2  1  1  .  .  1  2  1  2

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

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 -1  1 -1  1  1  1 -1  1
X.4     1  1 -1 -1  1 -1  1  1  1
X.5     2  .  .  . -2  .  2  .  2
X.6     4 -2  .  .  .  .  1  1 -2
X.7     4  . -2  .  .  1 -2  .  1
X.8     4  .  2  .  . -1 -2  .  1
X.9     4  2  .  .  .  .  1 -1 -2