Properties

Label 16T40
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $OD_{16}.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $40$
Group :  $OD_{16}.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,11,3,13,2,12,4,14)(5,15,7,10,6,16,8,9), (1,15,4,10,2,16,3,9)(5,12,8,14,6,11,7,13)
$|\Aut(F/K)|$:  $8$

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:  $C_2^2:C_4$

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

32T25

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 8]
Character table:   
      2  5  4  5  4  4  3  3  3  3  3  3

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

X.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
X.3      1  1  1  1  1 -1 -1  1 -1 -1  1
X.4      1  1  1  1  1  1  1 -1 -1 -1 -1
X.5      1  1  1 -1 -1 -1  1  A  A -A -A
X.6      1  1  1 -1 -1 -1  1 -A -A  A  A
X.7      1  1  1 -1 -1  1 -1  A -A  A -A
X.8      1  1  1 -1 -1  1 -1 -A  A -A  A
X.9      2 -2  2 -2  2  .  .  .  .  .  .
X.10     2 -2  2  2 -2  .  .  .  .  .  .
X.11     4  . -4  .  .  .  .  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i