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Properties

Label 16T19
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4 \times D_4$

Related objects

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Group action invariants

Degree $n$:  $16$
Transitive number $t$:  $19$
Group:  $C_4 \times D_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $2$
$|\Aut(F/K)|$:  $8$
Generators:  (1,11,2,12)(3,9,4,10)(5,7,6,8)(13,15,14,16), (1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14), (1,13,16,11)(2,14,15,12)(3,7,5,9)(4,8,6,10)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4\times C_2$, $Q_8:C_2$

Low degree siblings

16T19 x 3, 32T5

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [32, 25]
Character table:    
      2  5  4  5  4  4  5  4  5  5  5  4  4  4  4  4  4  4  4  5  5

        1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 2d 4g 2e 4h 4i 4j 4k 4l 2f 2g
     2P 1a 1a 1a 1a 2b 2b 2b 2b 2b 2b 1a 2f 1a 2f 2g 2b 2g 2b 1a 1a
     3P 1a 2a 2b 2c 4c 4d 4a 4b 4f 4e 2d 4g 2e 4h 4k 4l 4i 4j 2f 2g

X.1      1  1  1  1  1  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  1 -1  1 -1  1  1
X.3      1 -1  1 -1 -1  1 -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 -1  1 -1  1  1  1
X.5      1 -1  1 -1  1 -1  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  1  1  1  1  1  1
X.7      1  1  1  1 -1 -1 -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 -1 -1 -1 -1  1  1
X.9      1 -1 -1  1  A -A -A  A  A -A -1  1  1 -1 -A  A  A -A  1 -1
X.10     1 -1 -1  1 -A  A  A -A -A  A -1  1  1 -1  A -A -A  A  1 -1
X.11     1 -1 -1  1  A -A -A  A  A -A  1 -1 -1  1  A -A -A  A  1 -1
X.12     1 -1 -1  1 -A  A  A -A -A  A  1 -1 -1  1 -A  A  A -A  1 -1
X.13     1  1 -1 -1  A  A -A -A -A  A -1 -1  1  1 -A -A  A  A  1 -1
X.14     1  1 -1 -1 -A -A  A  A  A -A -1 -1  1  1  A  A -A -A  1 -1
X.15     1  1 -1 -1  A  A -A -A -A  A  1  1 -1 -1  A  A -A -A  1 -1
X.16     1  1 -1 -1 -A -A  A  A  A -A  1  1 -1 -1 -A -A  A  A  1 -1
X.17     2  .  2  .  . -2  . -2  2  2  .  .  .  .  .  .  .  . -2 -2
X.18     2  .  2  .  .  2  .  2 -2 -2  .  .  .  .  .  .  .  . -2 -2
X.19     2  . -2  .  .  B  . -B  B -B  .  .  .  .  .  .  .  . -2  2
X.20     2  . -2  .  . -B  .  B -B  B  .  .  .  .  .  .  .  . -2  2

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