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Properties

Label 16T7
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_8$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $7$
Group :  $D_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,14,2,13)(3,8,4,7)(5,10,6,9)(11,16,12,15), (1,15,2,16)(3,5,4,6)(7,10,8,9)(11,14,12,13), (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13)
$|\Aut(F/K)|$:  $16$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$, $Q_8$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8$ x 2

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 12]
Character table:    
      2  4  4  3  4  4  3  3  3  3  3

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

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 -2  .  2 -2  .  .  .  .  .
X.10     2 -2  . -2  2  .  .  .  .  .