↑

Properties

Label 16T14
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_{16}$

Related objects

Learn more about

Group action invariants

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

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: $C_2$ x 3

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

Degree 8: $D_4$

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 9]
Character table:    
     2  4  4  3  3  3  2  2

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

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

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