Properties

Label 32T26
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_8:C_2$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $26$
Group :  $D_8:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,19)(14,20)(15,18)(16,17)(21,26)(22,25)(23,27)(24,28)(29,32)(30,31), (1,27,29,24,4,26,31,22)(2,28,30,23,3,25,32,21)(5,14,10,19,7,16,11,18)(6,13,9,20,8,15,12,17), (1,16)(2,15)(3,13)(4,14)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(17,32)(18,31)(19,29)(20,30)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T44 x 2, 16T47

Low degree siblings

16T44 x 2, 16T47

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 42]
Character table:   
      2  5  3  5  3  5  5  4  4  3  4  3  4  4  4

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  .  2  . -2 -2  2  .  .  .  .  .  . -2
X.10     2  .  2  .  2  2 -2  .  .  .  .  .  . -2
X.11     2  . -2  .  A -A  .  B  . -B  .  C -C  .
X.12     2  . -2  .  A -A  . -B  .  B  . -C  C  .
X.13     2  . -2  . -A  A  .  B  . -B  . -C  C  .
X.14     2  . -2  . -A  A  . -B  .  B  .  C -C  .

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