Properties

Label 16T1036
Order \(672\)
n \(16\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1036$
CHM label :  $t16n1036$
Parity:  $-1$
Primitive:  No
Generators:   ( 1, 4,13, 5, 7,16,11, 9, 2, 3,14, 6, 8,15,12,10), ( 1, 8,16, 5,11, 4,10)( 2, 7,15, 6,12, 3, 9)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1
336: 8T43

Subfields

Degree 2: None

Degree 4: None

Degree 8: $\PGL(2,7)$

Low degree siblings

16T1036b
A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $56$ $2$ $( 3, 4)( 5, 8)( 6, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 6, 6, 2, 1, 1 $ $56$ $6$ $( 3, 4)( 5, 9,15, 8,14,11)( 6,10,16, 7,13,12)$
$ 6, 6, 2, 1, 1 $ $56$ $6$ $( 1, 2)( 5,10,15, 7,14,12)( 6, 9,16, 8,13,11)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $56$ $3$ $( 5,14,15)( 6,13,16)( 7,10,12)( 8, 9,11)$
$ 6, 6, 2, 2 $ $56$ $6$ $( 1, 2)( 3, 4)( 5,13,15, 6,14,16)( 7, 9,12, 8,10,11)$
$ 7, 7, 1, 1 $ $48$ $7$ $( 3, 6,13,11,16, 9, 8)( 4, 5,14,12,15,10, 7)$
$ 14, 2 $ $48$ $14$ $( 1, 2)( 3, 5,13,12,16,10, 8, 4, 6,14,11,15, 9, 7)$
$ 4, 4, 4, 4 $ $42$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,16,10,15)(11,13,12,14)$
$ 16 $ $42$ $16$ $( 1, 4, 6,11, 8,14,16, 9, 2, 3, 5,12, 7,13,15,10)$
$ 16 $ $42$ $16$ $( 1, 3, 6,12, 8,13,16,10, 2, 4, 5,11, 7,14,15, 9)$
$ 8, 8 $ $42$ $8$ $( 1, 4, 5,14, 2, 3, 6,13)( 7,11,10,15, 8,12, 9,16)$
$ 8, 8 $ $42$ $8$ $( 1, 3, 5,13, 2, 4, 6,14)( 7,12,10,16, 8,11, 9,15)$
$ 16 $ $42$ $16$ $( 1, 4, 6,15,12,13, 9, 8, 2, 3, 5,16,11,14,10, 7)$
$ 16 $ $42$ $16$ $( 1, 3, 6,16,12,14, 9, 7, 2, 4, 5,15,11,13,10, 8)$

Group invariants

Order:  $672=2^{5} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
Character table:  
      2  5  5  2  2  2  2  2  1   1  4   4   4  4  4   4   4
      3  1  1  1  1  1  1  1  .   .  .   .   .  .  .   .   .
      7  1  1  .  .  .  .  .  1   1  .   .   .  .  .   .   .

        1a 2a 2b 6a 6b 3a 6c 7a 14a 4a 16a 16b 8a 8b 16c 16d
     2P 1a 1a 1a 3a 3a 3a 3a 7a  7a 2a  8a  8a 4a 4a  8b  8b
     3P 1a 2a 2b 2b 2b 1a 2a 7a 14a 4a 16d 16c 8b 8a 16a 16b
     5P 1a 2a 2b 6b 6a 3a 6c 7a 14a 4a 16d 16c 8b 8a 16a 16b
     7P 1a 2a 2b 6a 6b 3a 6c 1a  2a 4a 16a 16b 8a 8b 16c 16d
    11P 1a 2a 2b 6b 6a 3a 6c 7a 14a 4a 16c 16d 8b 8a 16b 16a
    13P 1a 2a 2b 6a 6b 3a 6c 7a 14a 4a 16c 16d 8b 8a 16b 16a

X.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
X.3      6  6  .  .  .  .  . -1  -1 -2   .   .  2  2   .   .
X.4      6 -6  .  .  .  .  . -1   1  .   B  -B  D -D  -C   C
X.5      6 -6  .  .  .  .  . -1   1  .   C  -C -D  D   B  -B
X.6      6 -6  .  .  .  .  . -1   1  .  -C   C -D  D  -B   B
X.7      6 -6  .  .  .  .  . -1   1  .  -B   B  D -D   C  -C
X.8      6  6  .  .  .  .  . -1  -1  2   D   D  .  .  -D  -D
X.9      6  6  .  .  .  .  . -1  -1  2  -D  -D  .  .   D   D
X.10     7  7 -1 -1 -1  1  1  .   . -1   1   1 -1 -1   1   1
X.11     7  7  1  1  1  1  1  .   . -1  -1  -1 -1 -1  -1  -1
X.12     8  8 -2  1  1 -1 -1  1   1  .   .   .  .  .   .   .
X.13     8  8  2 -1 -1 -1 -1  1   1  .   .   .  .  .   .   .
X.14     8 -8  .  .  .  2 -2  1  -1  .   .   .  .  .   .   .
X.15     8 -8  .  A -A -1  1  1  -1  .   .   .  .  .   .   .
X.16     8 -8  . -A  A -1  1  1  -1  .   .   .  .  .   .   .

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
B = -E(16)-E(16)^7
C = -E(16)^3-E(16)^5
D = -E(8)+E(8)^3
  = -Sqrt(2) = -r2