Properties

Label 8T11
Degree $8$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8:C_2$

Related objects

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Group action invariants

Degree $n$:  $8$
Transitive number $t$:  $11$
Group:  $Q_8:C_2$
CHM label:  $1/2[2^{3}]E(4)=Q_{8}:2$
Parity:  $1$
Primitive:  no
Nilpotency class:  $2$
$|\Aut(F/K)|$:  $4$
Generators:  (1,3,5,7)(2,4,6,8), (1,5)(3,7), (1,4,5,8)(2,3,6,7)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T11 x 2, 16T11

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [16, 13]
Character table:   
      2  4  3  3  3  4  3  3  3  4  4

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

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

A = -2*E(4)
  = -2*Sqrt(-1) = -2i