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Properties

Label 16T163
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2.SD_{16}$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $163$
Group :  $C_2^2.SD_{16}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2)(5,6)(7,14,11,9)(8,13,12,10), (1,7,4,9,5,12,15,13)(2,8,3,10,6,11,16,14)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
32:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$, $C_4\wr C_2$, $C_2^3 : C_4 $

Low degree siblings

16T163, 32T174, 32T283, 32T319

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 8]
Character table:    
      2  6  5   4   4  6  5  5  4   4   4  6  6  3  3  4  4  4  4  5

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

X.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
X.3      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
X.5      1 -1   A  -A  1 -1 -1  1  -A   A  1  1 -1  1 -A  A -A  A -1
X.6      1 -1  -A   A  1 -1 -1  1   A  -A  1  1 -1  1  A -A  A -A -1
X.7      1 -1   A  -A  1 -1 -1  1  -A   A  1  1  1 -1  A -A  A -A -1
X.8      1 -1  -A   A  1 -1 -1  1   A  -A  1  1  1 -1 -A  A -A  A -1
X.9      2  2   .   .  2  2 -2 -2   .   .  2  2  .  .  .  .  .  . -2
X.10     2 -2   .   .  2 -2  2 -2   .   .  2  2  .  .  .  .  .  .  2
X.11     2 -2   .   . -2  2  .  .   .   . -2  2  .  .  D -D -D  D  .
X.12     2 -2   .   . -2  2  .  .   .   . -2  2  .  . -D  D  D -D  .
X.13     2  2   .   . -2 -2  .  .   .   . -2  2  .  .  E  E -E -E  .
X.14     2  2   .   . -2 -2  .  .   .   . -2  2  .  . -E -E  E  E  .
X.15     2  .   B  /B  2  .  C  . -/B  -B -2 -2  .  .  .  .  .  . -C
X.16     2  .  /B   B  2  . -C  .  -B -/B -2 -2  .  .  .  .  .  .  C
X.17     2  . -/B  -B  2  . -C  .   B  /B -2 -2  .  .  .  .  .  .  C
X.18     2  .  -B -/B  2  .  C  .  /B   B -2 -2  .  .  .  .  .  . -C
X.19     4  .   .   . -4  .  .  .   .   .  4 -4  .  .  .  .  .  .  .

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