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Properties

Label 8T33
Order \(96\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4:C_6$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $33$
Group :  $C_2^4:C_6$
CHM label :  $E(8):A_{4}=[1/3.A(4)^{2}]2=E(4):6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (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:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

8T33, 12T58 x 2, 12T59 x 2, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389

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, 2, 1, 1, 1, 1 $ $6$ $2$ $(4,5)(6,7)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,3,8)(5,7,6)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,8,3)(5,6,7)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 6, 2 $ $16$ $6$ $(1,4)(2,5,3,7,8,6)$
$ 6, 2 $ $16$ $6$ $(1,4)(2,6,8,7,3,5)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $12$ $4$ $(1,4,2,7)(3,6,8,5)$

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 70]
Character table:    
      2  5  4  1  1  4  5   1   1  3  3
      3  1  .  1  1  .  .   1   1  1  .

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

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

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