Properties

Label 32T31
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_{16}$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $31$
Group :  $D_{16}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,6)(2,5)(3,7)(4,8)(9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28)(16,27)(17,22)(18,21)(19,23)(20,24), (1,11)(2,12)(3,10)(4,9)(5,7)(6,8)(13,30)(14,29)(15,32)(16,31)(17,25)(18,26)(19,28)(20,27)(21,22)(23,24)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
16:  $D_{8}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $D_{8}$ x 2

Degree 16: $D_{8}$, $D_{16}$ x 2

Low degree siblings

16T56 x 2

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 2)( 3, 4)( 5,29)( 6,30)( 7,31)( 8,32)( 9,28)(10,27)(11,26)(12,25)(13,24) (14,23)(15,22)(16,21)(17,20)(18,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
$ 16, 16 $ $2$ $16$ $( 1, 5, 9,15,17,24,26,31, 3, 8,12,14,19,21,27,30)( 2, 6,10,16,18,23,25,32, 4, 7,11,13,20,22,28,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28) (16,27)(17,22)(18,21)(19,23)(20,24)$
$ 16, 16 $ $2$ $16$ $( 1, 8, 9,14,17,21,26,30, 3, 5,12,15,19,24,27,31)( 2, 7,10,13,18,22,25,29, 4, 6,11,16,20,23,28,32)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 9,17,26, 3,12,19,27)( 2,10,18,25, 4,11,20,28)( 5,15,24,31, 8,14,21,30) ( 6,16,23,32, 7,13,22,29)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,12,17,27, 3, 9,19,26)( 2,11,18,28, 4,10,20,25)( 5,14,24,30, 8,15,21,31) ( 6,13,23,29, 7,16,22,32)$
$ 16, 16 $ $2$ $16$ $( 1,14,26, 5,19,31, 9,21, 3,15,27, 8,17,30,12,24)( 2,13,25, 6,20,32,10,22, 4, 16,28, 7,18,29,11,23)$
$ 16, 16 $ $2$ $16$ $( 1,15,26, 8,19,30, 9,24, 3,14,27, 5,17,31,12,21)( 2,16,25, 7,20,29,10,23, 4, 13,28, 6,18,32,11,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,17, 3,19)( 2,18, 4,20)( 5,24, 8,21)( 6,23, 7,22)( 9,26,12,27)(10,25,11,28) (13,29,16,32)(14,30,15,31)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 18]
Character table:   
      2  5  2  5   4  2   4  4  4   4   4  4

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

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      2  .  2   .  .   . -2 -2   .   .  2
X.6      2  .  2   A  .   A  .  .  -A  -A -2
X.7      2  .  2  -A  .  -A  .  .   A   A -2
X.8      2  . -2   B  .  -B -A  A  -C   C  .
X.9      2  . -2   C  .  -C  A -A   B  -B  .
X.10     2  . -2  -C  .   C  A -A  -B   B  .
X.11     2  . -2  -B  .   B -A  A   C  -C  .

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