Properties

Label 16T16
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_8:C_2):C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $16$
Group :  $(C_8:C_2):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,6,2,5)(3,8,4,7)(9,14,10,13)(11,16,12,15), (1,14,6,9,2,13,5,10)(3,15,8,12,4,16,7,11), (1,16)(2,15)(3,10)(4,9)(5,12)(6,11)(7,13)(8,14)
$|\Aut(F/K)|$:  $8$

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:  $C_4\times C_2$ x 6, $C_2^3$
16:  $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$

Low degree siblings

16T16 x 2, 32T2

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 38]
Character table:   
      2  5  4  5  4   5   5  4  5  5   5  4   5  4  4  4  4  4  4  4  4

        1a 2a 2b 8a  8b  8c 4a 4b 4c  8d 8e  8f 8g 8h 4d 2c 8i 8j 2d 4e
     2P 1a 1a 1a 4c  4c  4c 2b 2b 2b  4b 4b  4b 4b 4c 2b 1a 4c 4b 1a 2b
     3P 1a 2a 2b 8e  8f  8d 4a 4c 4b  8c 8a  8b 8i 8j 4d 2c 8g 8h 2d 4e
     5P 1a 2a 2b 8a  8c  8b 4a 4b 4c  8f 8e  8d 8g 8h 4d 2c 8i 8j 2d 4e
     7P 1a 2a 2b 8e  8d  8f 4a 4c 4b  8b 8a  8c 8i 8j 4d 2c 8g 8h 2d 4e

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  A  -A  -A  1 -1 -1   A -A   A  A -A -1  1 -A  A  1 -1
X.10     1 -1  1 -A   A   A  1 -1 -1  -A  A  -A -A  A -1  1  A -A  1 -1
X.11     1 -1  1  A  -A  -A  1 -1 -1   A -A   A -A  A  1 -1  A -A -1  1
X.12     1 -1  1 -A   A   A  1 -1 -1  -A  A  -A  A -A  1 -1 -A  A -1  1
X.13     1  1  1  A   A   A -1 -1 -1  -A -A  -A  A  A -1 -1 -A -A  1  1
X.14     1  1  1 -A  -A  -A -1 -1 -1   A  A   A -A -A -1 -1  A  A  1  1
X.15     1  1  1  A   A   A -1 -1 -1  -A -A  -A -A -A  1  1  A  A -1 -1
X.16     1  1  1 -A  -A  -A -1 -1 -1   A  A   A  A  A  1  1 -A -A -1 -1
X.17     2  . -2  .   B  -B  .  C -C  /B  . -/B  .  .  .  .  .  .  .  .
X.18     2  . -2  . -/B  /B  . -C  C  -B  .   B  .  .  .  .  .  .  .  .
X.19     2  . -2  .  /B -/B  . -C  C   B  .  -B  .  .  .  .  .  .  .  .
X.20     2  . -2  .  -B   B  .  C -C -/B  .  /B  .  .  .  .  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(8)
C = -2*E(4)
  = -2*Sqrt(-1) = -2i