Properties

Label 8T46
Order \(576\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4^2:C_4$

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $46$
Group :  $A_4^2:C_4$
CHM label :  $1/2[S(4)^{2}]2$
Parity:  $-1$
Primitive:  No
Generators:   (1,4,3,6)(2,5)(7,8), (1,4,2,6,8,7,3,5)
$|\textrm{Aut}(F/K)|$:  $1$
Low degree resolvents:  
2: 2T1
4: 4T1
36: 6T10

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T160, 12T162, 16T1030, 16T1031, 18T182, 18T184
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 $ $6$ $2$ $(1,3)(2,8)$
$ 2, 2, 2, 2 $ $9$ $2$ $(1,3)(2,8)(4,7)(5,6)$
$ 3, 1, 1, 1, 1, 1 $ $16$ $3$ $(2,8,3)$
$ 3, 2, 2, 1 $ $48$ $6$ $(2,8,3)(4,7)(5,6)$
$ 3, 3, 1, 1 $ $64$ $3$ $(2,8,3)(5,7,6)$
$ 2, 2, 1, 1, 1, 1 $ $36$ $2$ $(3,8)(6,7)$
$ 4, 2, 1, 1 $ $72$ $4$ $(1,3,2,8)(6,7)$
$ 4, 4 $ $36$ $4$ $(1,3,2,8)(4,7,5,6)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,7,3,6)(4,8)$
$ 8 $ $72$ $8$ $(1,5,3,6,8,4,2,7)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,6)(3,7,8,4)$
$ 8 $ $72$ $8$ $(1,5,3,7,2,6,8,4)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  6  5  6  2  2  .  4  3  4  3  3  3  3
      3  2  1  .  2  1  2  .  .  .  .  .  .  .

        1a 2a 2b 3a 6a 3b 2c 4a 4b 4c 8a 4d 8b
     2P 1a 1a 1a 3a 3a 3b 1a 2a 2b 2c 4b 2c 4b
     3P 1a 2a 2b 1a 2a 1a 2c 4a 4b 4d 8b 4c 8a
     5P 1a 2a 2b 3a 6a 3b 2c 4a 4b 4c 8a 4d 8b
     7P 1a 2a 2b 3a 6a 3b 2c 4a 4b 4d 8b 4c 8a

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
X.3      1  1  1  1  1  1 -1 -1 -1  A  A -A -A
X.4      1  1  1  1  1  1 -1 -1 -1 -A -A  A  A
X.5      4  4  4 -2 -2  1  .  .  .  .  .  .  .
X.6      4  4  4  1  1 -2  .  .  .  .  .  .  .
X.7      6  2 -2  3 -1  . -2  .  2  .  .  .  .
X.8      6  2 -2  3 -1  .  2  . -2  .  .  .  .
X.9      9 -3  1  .  .  .  1 -1  1 -1  1 -1  1
X.10     9 -3  1  .  .  .  1 -1  1  1 -1  1 -1
X.11     9 -3  1  .  .  . -1  1 -1  A -A -A  A
X.12     9 -3  1  .  .  . -1  1 -1 -A  A  A -A
X.13    12  4 -4 -3  1  .  .  .  .  .  .  .  .

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