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Properties

Label 16T12
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $QD_{16}$

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Group action invariants

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

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: $D_4$, $QD_{16}$

Low degree siblings

8T8

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 8]
Character table:    
     2  4  2  3  2  3  4  3

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

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

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2