↑

Properties

Label 32T1
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times OD_{16}$

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $C_4\times C_2$ x 6, $C_2^3$
16:  $C_8:C_2$ x 2, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $C_8:C_2$ x 2

Degree 16: $C_4\times C_2^2$, $C_8: C_2$ x 2, $C_2 \times (C_8:C_2)$ x 2

Low degree siblings

16T15 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5,22)( 6,21)( 7,23)( 8,24)( 9,10)(11,12)(13,29)(14,30)(15,32) (16,31)(17,18)(19,20)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,24)( 6,23)( 7,21)( 8,22)( 9,12)(10,11)(13,32)(14,31)(15,29) (16,30)(17,19)(18,20)(25,27)(26,28)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 5, 9,13,20,21,27,30)( 2, 6,10,14,19,22,28,29)( 3, 8,11,15,17,23,26,31) ( 4, 7,12,16,18,24,25,32)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 6,27,29,20,22, 9,14)( 2, 5,28,30,19,21,10,13)( 3, 7,26,32,17,24,11,16) ( 4, 8,25,31,18,23,12,15)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 7,27,32,20,24, 9,16)( 2, 8,28,31,19,23,10,15)( 3, 6,26,29,17,22,11,14) ( 4, 5,25,30,18,21,12,13)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 8, 9,15,20,23,27,31)( 2, 7,10,16,19,24,28,32)( 3, 5,11,13,17,21,26,30) ( 4, 6,12,14,18,22,25,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 9,20,27)( 2,10,19,28)( 3,11,17,26)( 4,12,18,25)( 5,13,21,30)( 6,14,22,29) ( 7,16,24,32)( 8,15,23,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,10,20,28)( 2, 9,19,27)( 3,12,17,25)( 4,11,18,26)( 5,29,21,14)( 6,30,22,13) ( 7,31,24,15)( 8,32,23,16)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11,20,26)( 2,12,19,25)( 3, 9,17,27)( 4,10,18,28)( 5,15,21,31)( 6,16,22,32) ( 7,14,24,29)( 8,13,23,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,12,20,25)( 2,11,19,26)( 3,10,17,28)( 4, 9,18,27)( 5,32,21,16)( 6,31,22,15) ( 7,30,24,13)( 8,29,23,14)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,13,27, 5,20,30, 9,21)( 2,14,28, 6,19,29,10,22)( 3,15,26, 8,17,31,11,23) ( 4,16,25, 7,18,32,12,24)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,14, 9,22,20,29,27, 6)( 2,13,10,21,19,30,28, 5)( 3,16,11,24,17,32,26, 7) ( 4,15,12,23,18,31,25, 8)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,15,27, 8,20,31, 9,23)( 2,16,28, 7,19,32,10,24)( 3,13,26, 5,17,30,11,21) ( 4,14,25, 6,18,29,12,22)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,16, 9,24,20,32,27, 7)( 2,15,10,23,19,31,28, 8)( 3,14,11,22,17,29,26, 6) ( 4,13,12,21,18,30,25, 5)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,23)( 6,24)( 7,22)( 8,21)( 9,26)(10,25)(11,27) (12,28)(13,31)(14,32)(15,30)(16,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,24)( 8,23)( 9,27)(10,28)(11,26) (12,25)(13,30)(14,29)(15,31)(16,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,26,20,11)( 2,25,19,12)( 3,27,17, 9)( 4,28,18,10)( 5,31,21,15)( 6,32,22,16) ( 7,29,24,14)( 8,30,23,13)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,27,20, 9)( 2,28,19,10)( 3,26,17,11)( 4,25,18,12)( 5,30,21,13)( 6,29,22,14) ( 7,32,24,16)( 8,31,23,15)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 37]
Character table:    
      2  5  4  5  4  4  4  4  4  5  4  5  4  4  4  4  4  5  5  5  5

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

X.1      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  1
X.3      1 -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  1
X.5      1 -1  1 -1  1 -1 -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  1  1 -1  1 -1  1
X.7      1  1 -1 -1  1  1 -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 -1 -1  1  1  1  1
X.9      1 -1 -1  1  A -A  A -A -1  1  1 -1 -A  A  A -A -1  1  1 -1
X.10     1 -1 -1  1 -A  A -A  A -1  1  1 -1  A -A -A  A -1  1  1 -1
X.11     1 -1  1 -1  A -A -A  A -1  1 -1  1 -A  A -A  A  1  1 -1 -1
X.12     1 -1  1 -1 -A  A  A -A -1  1 -1  1  A -A  A -A  1  1 -1 -1
X.13     1  1 -1 -1  A  A -A -A -1 -1  1  1 -A -A  A  A -1  1  1 -1
X.14     1  1 -1 -1 -A -A  A  A -1 -1  1  1  A  A -A -A -1  1  1 -1
X.15     1  1  1  1  A  A  A  A -1 -1 -1 -1 -A -A -A -A  1  1 -1 -1
X.16     1  1  1  1 -A -A -A -A -1 -1 -1 -1  A  A  A  A  1  1 -1 -1
X.17     2  . -2  .  .  .  .  .  B  . -B  .  .  .  .  .  2 -2  B -B
X.18     2  . -2  .  .  .  .  . -B  .  B  .  .  .  .  .  2 -2 -B  B
X.19     2  .  2  .  .  .  .  .  B  .  B  .  .  .  .  . -2 -2 -B -B
X.20     2  .  2  .  .  .  .  . -B  . -B  .  .  .  .  . -2 -2  B  B

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i