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Properties

Label 16T6
Degree $16$
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 $n$:  $16$
Transitive number $t$:  $6$
Group:  $C_8: C_2$
Parity:  $1$
Primitive:  no
Nilpotency class:  $2$
$|\Aut(F/K)|$:  $16$
Generators:  (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,3,5,8,10,12,14,15)(2,4,6,7,9,11,13,16)

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: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $C_8:C_2$

Low degree siblings

8T7

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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 8,10,12,14,15)( 2, 4, 6, 7, 9,11,13,16)$
$ 8, 8 $ $2$ $8$ $( 1, 4,14,16,10,11, 5, 7)( 2, 3,13,15, 9,12, 6, 8)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,16,12, 7)( 4,15,11, 8)$
$ 8, 8 $ $2$ $8$ $( 1, 7, 5,11,10,16,14, 4)( 2, 8, 6,12, 9,15,13, 3)$
$ 8, 8 $ $2$ $8$ $( 1, 8,14, 3,10,15, 5,12)( 2, 7,13, 4, 9,16, 6,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,14,10, 5)( 2,13, 9, 6)( 3,15,12, 8)( 4,16,11, 7)$

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