Properties

Label 32T38
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\times Q_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $38$
Group :  $C_4\times Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,27,2,28)(3,25,4,26)(5,14,6,13)(7,16,8,15)(9,17,10,18)(11,20,12,19)(21,31,22,32)(23,29,24,30), (1,16,12,21)(2,15,11,22)(3,13,9,23)(4,14,10,24)(5,17,29,25)(6,18,30,26)(7,20,32,28)(8,19,31,27), (1,9,2,10)(3,11,4,12)(5,31,6,32)(7,29,8,30)(13,22,14,21)(15,24,16,23)(17,27,18,28)(19,26,20,25)
$|\Aut(F/K)|$:  $32$

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, $C_2^3$, $Q_8$ x 2
16:  $Q_8:C_2$, $C_4\times C_2^2$, $D_8$

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, $C_2^3$, $Q_8$ x 2, $Q_8:C_2$ x 3

Degree 16: $C_4\times C_2^2$, $D_8$, $Q_8 : C_2$

Low degree siblings

There are no siblings 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 5,11,30)( 2, 6,12,29)( 3, 7,10,31)( 4, 8, 9,32)(13,20,24,27)(14,19,23,28) (15,18,21,25)(16,17,22,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,11,31)( 2, 8,12,32)( 3, 6,10,29)( 4, 5, 9,30)(13,18,24,25)(14,17,23,26) (15,19,21,28)(16,20,22,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,31, 6,32)( 7,29, 8,30)(13,22,14,21)(15,24,16,23) (17,27,18,28)(19,26,20,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,31)( 8,32)(13,24)(14,23)(15,21) (16,22)(17,26)(18,25)(19,28)(20,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,32)( 8,31)(13,23)(14,24)(15,22) (16,21)(17,25)(18,26)(19,27)(20,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13,11,24)( 2,14,12,23)( 3,15,10,21)( 4,16, 9,22)( 5,19,30,28)( 6,20,29,27) ( 7,17,31,26)( 8,18,32,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,15,12,22)( 2,16,11,21)( 3,14, 9,24)( 4,13,10,23)( 5,18,29,26)( 6,17,30,25) ( 7,19,32,27)( 8,20,31,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,16,12,21)( 2,15,11,22)( 3,13, 9,23)( 4,14,10,24)( 5,17,29,25)( 6,18,30,26) ( 7,20,32,28)( 8,19,31,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,17, 2,18)( 3,20, 4,19)( 5,22, 6,21)( 7,24, 8,23)( 9,28,10,27)(11,26,12,25) (13,32,14,31)(15,30,16,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,19, 2,20)( 3,17, 4,18)( 5,24, 6,23)( 7,21, 8,22)( 9,25,10,26)(11,28,12,27) (13,29,14,30)(15,32,16,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,21,12,16)( 2,22,11,15)( 3,23, 9,13)( 4,24,10,14)( 5,25,29,17)( 6,26,30,18) ( 7,28,32,20)( 8,27,31,19)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,22,12,15)( 2,21,11,16)( 3,24, 9,14)( 4,23,10,13)( 5,26,29,18)( 6,25,30,17) ( 7,27,32,19)( 8,28,31,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23,11,14)( 2,24,12,13)( 3,22,10,16)( 4,21, 9,15)( 5,27,30,20)( 6,28,29,19) ( 7,25,31,18)( 8,26,32,17)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,25, 2,26)( 3,28, 4,27)( 5,15, 6,16)( 7,14, 8,13)( 9,20,10,19)(11,18,12,17) (21,29,22,30)(23,32,24,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,27, 2,28)( 3,25, 4,26)( 5,14, 6,13)( 7,16, 8,15)( 9,17,10,18)(11,20,12,19) (21,31,22,32)(23,29,24,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,29,11, 6)( 2,30,12, 5)( 3,32,10, 8)( 4,31, 9, 7)(13,28,24,19)(14,27,23,20) (15,26,21,17)(16,25,22,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,31,11, 7)( 2,32,12, 8)( 3,29,10, 6)( 4,30, 9, 5)(13,25,24,18)(14,26,23,17) (15,28,21,19)(16,27,22,20)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 26]
Character table:   
      2  5  5  4  4  4  4  5  5  4  5  5  4  4  5  5  4  4  4  4  4

        1a 2a 4a 4b 4c 4d 2b 2c 4e 4f 4g 4h 4i 4j 4k 4l 4m 4n 4o 4p
     2P 1a 1a 2a 2b 2b 2a 1a 1a 2b 2c 2c 2a 2a 2c 2c 2b 2a 2a 2b 2b
     3P 1a 2a 4a 4o 4p 4d 2b 2c 4l 4k 4j 4h 4i 4g 4f 4e 4m 4n 4b 4c

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

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