Properties

Label 32T27
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times SD_{16}$

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Group action invariants

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

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:  $QD_{16}$ x 2, $D_4\times C_2$

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, $QD_{16}$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, $QD_{16}$ x 2, 16T48 x 2

Low degree siblings

16T48 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,14)( 6,13)( 7,16)( 8,15)( 9,25)(10,26)(11,27)(12,28)(17,18) (19,20)(21,29)(22,30)(23,32)(24,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,28)(10,27)(11,26)(12,25)(17,19) (18,20)(21,31)(22,32)(23,30)(24,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,31)(30,32)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 5,12,13,17,23,27,31)( 2, 6,11,14,18,24,28,32)( 3, 7,10,15,19,21,25,30) ( 4, 8, 9,16,20,22,26,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 6,17,24)( 2, 5,18,23)( 3, 8,19,22)( 4, 7,20,21)( 9,30,26,15)(10,29,25,16) (11,31,28,13)(12,32,27,14)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 7,17,21)( 2, 8,18,22)( 3, 5,19,23)( 4, 6,20,24)( 9,32,26,14)(10,31,25,13) (11,29,28,16)(12,30,27,15)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 8,12,16,17,22,27,29)( 2, 7,11,15,18,21,28,30)( 3, 6,10,14,19,24,25,32) ( 4, 5, 9,13,20,23,26,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 9,17,26)( 2,10,18,25)( 3,11,19,28)( 4,12,20,27)( 5,16,23,29)( 6,15,24,30) ( 7,14,21,32)( 8,13,22,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,12,17,27)( 2,11,18,28)( 3,10,19,25)( 4, 9,20,26)( 5,13,23,31)( 6,14,24,32) ( 7,15,21,30)( 8,16,22,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,26)(10,25)(11,28) (12,27)(13,31)(14,32)(15,30)(16,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,22)( 6,21)( 7,24)( 8,23)( 9,27)(10,28)(11,25) (12,26)(13,29)(14,30)(15,32)(16,31)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,22,12,29,17, 8,27,16)( 2,21,11,30,18, 7,28,15)( 3,24,10,32,19, 6,25,14) ( 4,23, 9,31,20, 5,26,13)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,23,12,31,17, 5,27,13)( 2,24,11,32,18, 6,28,14)( 3,21,10,30,19, 7,25,15) ( 4,22, 9,29,20, 8,26,16)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 40]
Character table:   
      2  5  3  3  5  4  3  3  4  4  4  5  5  4  4

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

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  .  . -2  2  A -A
X.12     2  .  . -2 -A  .  .  A  .  . -2  2 -A  A
X.13     2  .  .  2  A  .  .  A  .  . -2 -2 -A -A
X.14     2  .  .  2 -A  .  . -A  .  . -2 -2  A  A

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2