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Properties

Label 8T6
Order \(16\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_{8}$

Related objects

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Group action invariants

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

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

8T6, 16T13

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 7]
Character table:    
     2  4  2  2  3  3  3  4

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

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  . -A -2
X.7     2  .  . -A  .  A -2

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2