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Properties

Label 32T14
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\wr C_2$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $14$
Group :  $C_4\wr C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,7,27,14)(2,8,28,13)(3,5,26,15)(4,6,25,16)(9,29,18,21)(10,30,17,22)(11,31,19,23)(12,32,20,24), (1,8,10,29,4,5,12,31)(2,7,9,30,3,6,11,32)(13,20,21,27,15,17,23,25)(14,19,22,28,16,18,24,26)
$|\Aut(F/K)|$:  $32$

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

Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2, $C_4\wr C_2$ x 2

Degree 16: $C_2^2 : C_4$, 16T28, 16T42

Low degree siblings

8T17 x 2, 16T28, 16T42

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 11]
Character table:    
      2  5  3  5  3   4   4  3  5  5  3   4   4  4  4

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

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  A  -A  -A  1 -1 -1 -A   A   A  1 -1
X.6      1 -1  1 -A   A   A  1 -1 -1  A  -A  -A  1 -1
X.7      1  1  1  A   A   A -1 -1 -1 -A  -A  -A  1 -1
X.8      1  1  1 -A  -A  -A -1 -1 -1  A   A   A  1 -1
X.9      2  .  2  .   .   .  . -2 -2  .   .   . -2  2
X.10     2  .  2  .   .   .  .  2  2  .   .   . -2 -2
X.11     2  . -2  .   B  -B  .  C -C  . -/B  /B  .  .
X.12     2  . -2  .  /B -/B  . -C  C  .  -B   B  .  .
X.13     2  . -2  . -/B  /B  . -C  C  .   B  -B  .  .
X.14     2  . -2  .  -B   B  .  C -C  .  /B -/B  .  .

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