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Properties

Label 8T38
Order \(192\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\wr A_4$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $38$
Group :  $C_2\wr A_4$
CHM label :  $[2^{4}]A(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,3)(4,5)(6,7), (1,2,3)(5,6,7), (4,8)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
96:  $C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Low degree siblings

8T38, 16T425, 16T427, 24T288 x 2, 24T425 x 2, 32T2185 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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(3,7)(4,8)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,3,4)(6,7,8)$
$ 6, 1, 1 $ $16$ $6$ $(2,3,4,6,7,8)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,4,3)(6,8,7)$
$ 6, 1, 1 $ $16$ $6$ $(2,4,7,6,8,3)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $24$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 3, 3, 2 $ $16$ $6$ $(1,2,3)(4,8)(5,6,7)$
$ 6, 2 $ $16$ $6$ $(1,2,3,5,6,7)(4,8)$
$ 3, 3, 2 $ $16$ $6$ $(1,2,4)(3,7)(5,6,8)$
$ 6, 2 $ $16$ $6$ $(1,2,4,5,6,8)(3,7)$
$ 4, 4 $ $12$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 201]
Character table:    
      2  6  4  5  2   2  2   2  4  4  3   2   2   2   2  4  6
      3  1  1  .  1   1  1   1  1  .  .   1   1   1   1  .  1

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

X.1      1  1  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   1  1  1
X.3      1 -1  1  A  -A /A -/A -1  1 -1 -/A  /A  -A   A  1  1
X.4      1 -1  1 /A -/A  A  -A -1  1 -1  -A   A -/A  /A  1  1
X.5      1  1  1  A   A /A  /A  1  1  1  /A  /A   A   A  1  1
X.6      1  1  1 /A  /A  A   A  1  1  1   A   A  /A  /A  1  1
X.7      3 -3  3  .   .  .   . -3 -1  1   .   .   .   . -1  3
X.8      3  3  3  .   .  .   .  3 -1 -1   .   .   .   . -1  3
X.9      4 -2  .  1  -1  1  -1  2  .  .   1  -1   1  -1  . -4
X.10     4  2  .  1   1  1   1 -2  .  .  -1  -1  -1  -1  . -4
X.11     4 -2  .  A  -A /A -/A  2  .  .  /A -/A   A  -A  . -4
X.12     4 -2  . /A -/A  A  -A  2  .  .   A  -A  /A -/A  . -4
X.13     4  2  .  A   A /A  /A -2  .  . -/A -/A  -A  -A  . -4
X.14     4  2  . /A  /A  A   A -2  .  .  -A  -A -/A -/A  . -4
X.15     6  . -2  .   .  .   .  . -2  .   .   .   .   .  2  6
X.16     6  . -2  .   .  .   .  .  2  .   .   .   .   . -2  6

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