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Properties

Label 8T4
Degree $8$
Order $8$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4$

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Group action invariants

Degree $n$:  $8$
Transitive number $t$:  $4$
Group:  $D_4$
CHM label:  $D_{8}(8)=[4]2$
Parity:  $1$
Primitive:  no
Nilpotency class:  $2$
$\card{\Aut(F/K)}$:  $8$
Generators:  (1,2,3,8)(4,5,6,7), (1,6)(2,5)(3,4)(7,8)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Low degree siblings

4T3 x 2

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

Group invariants

Order:  $8=2^{3}$
Cyclic:  no
Abelian:  no
Solvable:  yes
Label:  8.3
Character table:    
     2  3  2  3  2  2

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

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  .  .