Properties

Label 8T5
Order \(8\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_8$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $5$
Group :  $Q_8$
CHM label :  $Q_{8}(8)$
Parity:  $1$
Primitive:  No
Generators:  (1,2,3,8)(4,5,6,7), (1,7,3,5)(2,6,8,4)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $V_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $V_4$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 4, 4 $ $2$ $4$ $(1,2,3,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 4, 4 $ $2$ $4$ $(1,4,3,6)(2,7,8,5)$
$ 4, 4 $ $2$ $4$ $(1,5,3,7)(2,4,8,6)$

Group invariants

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

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

X.1     1  1  1  1  1
X.2     1 -1  1 -1  1
X.3     1 -1  1  1 -1
X.4     1  1  1 -1 -1
X.5     2  . -2  .  .