Properties

Label 16T11
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_8 : C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $11$
Group :  $Q_8 : C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,5,10,14)(2,6,9,13)(3,7,12,16)(4,8,11,15), (1,7,10,16)(2,8,9,15)(3,5,12,14)(4,6,11,13)
$|\Aut(F/K)|$:  $16$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 3

Low degree siblings

8T11 x 3

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 13]
Character table:   
      2  4  3  3  3  4  3  3  3  4  4

        1a 2a 2b 4a 4b 4c 4d 2c 2d 4e
     2P 1a 1a 1a 2d 2d 2d 2d 1a 1a 2d
     3P 1a 2a 2b 4a 4e 4c 4d 2c 2d 4b

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

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