Properties

Label 16T47
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_8:C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $47$
Group :  $D_8:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,13)(2,14)(3,12)(4,11)(5,9)(6,10)(7,15)(8,16), (1,8,6,4,2,7,5,3)(9,16,14,12,10,15,13,11), (1,10,2,9)(3,16,4,15)(5,13,6,14)(7,12,8,11)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

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

16T44 x 2, 32T26

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 42]
Character table:   
      2  5  4  5  4  4  4  4  4  5  5  3  3  3  3

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

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1  1 -1  1 -1 -1 -1  1  1 -1
X.3      1 -1  1 -1  1  1 -1  1 -1 -1  1 -1 -1  1
X.4      1 -1  1  1 -1 -1  1  1 -1 -1 -1  1 -1  1
X.5      1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1 -1
X.6      1  1  1 -1 -1 -1 -1  1  1  1 -1 -1  1  1
X.7      1  1  1 -1 -1 -1 -1  1  1  1  1  1 -1 -1
X.8      1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
X.9      2  2  2  .  .  .  . -2 -2 -2  .  .  .  .
X.10     2 -2  2  .  .  .  . -2  2  2  .  .  .  .
X.11     2  . -2  A  B -B -A  .  C -C  .  .  .  .
X.12     2  . -2  A -B  B -A  . -C  C  .  .  .  .
X.13     2  . -2 -A  B -B  A  . -C  C  .  .  .  .
X.14     2  . -2 -A -B  B  A  .  C -C  .  .  .  .

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
B = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
C = 2*E(4)
  = 2*Sqrt(-1) = 2i