Properties

Label 8T8
Degree $8$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $QD_{16}$

Related objects

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Group action invariants

Degree $n$:  $8$
Transitive number $t$:  $8$
Group:  $QD_{16}$
CHM label:  $2D_{8}(8)=[D(4)]2$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $3$
$|\Aut(F/K)|$:  $2$
Generators:  (1,2,3,4,5,6,7,8), (1,3)(2,6)(5,7)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

16T12

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [16, 8]
Character table:   
     2  4  2  3  2  3  4  3

       1a 2a 8a 4a 4b 2b 8b
    2P 1a 1a 4b 2b 2b 1a 4b
    3P 1a 2a 8a 4a 4b 2b 8b
    5P 1a 2a 8b 4a 4b 2b 8a
    7P 1a 2a 8b 4a 4b 2b 8a

X.1     1  1  1  1  1  1  1
X.2     1 -1 -1  1  1  1 -1
X.3     1 -1  1 -1  1  1  1
X.4     1  1 -1 -1  1  1 -1
X.5     2  .  .  . -2  2  .
X.6     2  .  A  .  . -2 -A
X.7     2  . -A  .  . -2  A

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2