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Properties

Label 8T7
Degree $8$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8:C_2$

Related objects

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Group action invariants

Degree $n$:  $8$
Transitive number $t$:  $7$
Group:  $C_8:C_2$
CHM label:  $1/2[2^{3}]4$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $2$
$|\Aut(F/K)|$:  $4$
Generators:  (1,2,3,4,5,6,7,8), (1,5)(3,7)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

16T6

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, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 8 $ $2$ $8$ $(1,2,3,4,5,6,7,8)$
$ 8 $ $2$ $8$ $(1,2,7,8,5,6,3,4)$
$ 4, 4 $ $1$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 4, 4 $ $2$ $4$ $(1,3,5,7)(2,8,6,4)$
$ 8 $ $2$ $8$ $(1,4,3,6,5,8,7,2)$
$ 8 $ $2$ $8$ $(1,4,7,2,5,8,3,6)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$
$ 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, 6]
Character table:    
      2  4  3  3  3  4  3  3  3  4  4

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

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  A -A -1  1  A -A  1 -1
X.6      1 -1 -A  A -1  1 -A  A  1 -1
X.7      1  1  A  A -1 -1 -A -A  1 -1
X.8      1  1 -A -A -1 -1  A  A  1 -1
X.9      2  .  .  .  B  .  .  . -2 -B
X.10     2  .  .  . -B  .  .  . -2  B

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i