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Properties

Label 8T42
Order \(288\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $42$
Group :  $A_4\wr C_2$
CHM label :  $[A(4)^{2}]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8), (1,2,3), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T126, 12T128, 12T129, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1 $ $8$ $3$ $(5,6,7)$
$ 3, 1, 1, 1, 1, 1 $ $8$ $3$ $(5,7,6)$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(4,5)(6,7)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,8)(5,6,7)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,3,8)(5,7,6)$
$ 3, 2, 2, 1 $ $24$ $6$ $(2,3,8)(4,5)(6,7)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,8,3)(5,6,7)$
$ 3, 2, 2, 1 $ $24$ $6$ $(2,8,3)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $9$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,4)(2,5)(3,7)(6,8)$
$ 6, 2 $ $48$ $6$ $(1,4)(2,5,3,7,8,6)$
$ 6, 2 $ $48$ $6$ $(1,4)(2,5,8,6,3,7)$
$ 4, 4 $ $36$ $4$ $(1,4,2,5)(3,7,8,6)$

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [288, 1025]
Character table:    
      2  5   2   2  4  .  1   2  1   2  5  3   1   1  3
      3  2   2   2  1  2  2   1  2   1  .  1   1   1  .

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3