Properties

Label 8T32
Order \(96\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $32$
CHM label :  $[2^{3}]A(4)$
Parity:  $1$
Primitive:  No
Generators:   (1,2)(3,8)(4,7)(5,6), (1,3,2)(4,5,6), (1,4,7)(3,8,6)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
3: 3T1
12: 4T4, 4T4, 4T4, 4T4, 4T4
48: 12T32

Subfields

Degree 2: None

Degree 4: $A_4$

Low degree siblings

8T32b, 8T32c
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(3,4)(7,8)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,3,7)(4,8,5)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,7,3)(4,5,8)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,7)(4,8)(5,6)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 6, 2 $ $16$ $6$ $(1,2,3,6,5,4)(7,8)$
$ 4, 4 $ $6$ $4$ $(1,2,6,5)(3,7,4,8)$
$ 4, 4 $ $6$ $4$ $(1,2,6,5)(3,8,4,7)$
$ 6, 2 $ $16$ $6$ $(1,2,7,6,5,8)(3,4)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  5  4  1  1  4  4   1  4  4   1  5
      3  1  .  1  1  .  .   1  .  .   1  1

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

X.1      1  1  1  1  1  1   1  1  1   1  1
X.2      1  1  A /A  1  1  /A  1  1   A  1
X.3      1  1 /A  A  1  1   A  1  1  /A  1
X.4      3  3  .  . -1 -1   . -1 -1   .  3
X.5      3 -1  .  .  3 -1   . -1 -1   .  3
X.6      3 -1  .  . -1  3   . -1 -1   .  3
X.7      3 -1  .  . -1 -1   . -1  3   .  3
X.8      3 -1  .  . -1 -1   .  3 -1   .  3
X.9      4  .  1  1  .  .  -1  .  .  -1 -4
X.10     4  .  A /A  .  . -/A  .  .  -A -4
X.11     4  . /A  A  .  .  -A  .  . -/A -4

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3