Properties

Label 8T11
Order \(16\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_8:C_2$

Related objects

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
Generators:   (1,3,5,7)(2,4,6,8), (1,4,5,8)(2,3,6,7), (1,5)(3,7)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:  
2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 8T3

Subfields

Degree 2: $C_2$, $C_2$, $C_2$

Degree 4: $V_4$

Low degree siblings

8T11b, 8T11c, 16T11
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