Properties

Label 32T47
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_8:C_4$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $47$
Group :  $C_8:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,24,11,16)(2,23,12,15)(3,22,9,14)(4,21,10,13)(5,28,29,18)(6,27,30,17)(7,26,31,20)(8,25,32,19), (1,8,12,31)(2,7,11,32)(3,6,10,29)(4,5,9,30)(13,20,22,25)(14,19,21,26)(15,17,24,28)(16,18,23,27)
$|\Aut(F/K)|$:  $32$

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:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $QD_{16}$ x 2, $C_4:C_4$

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 2

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$, $QD_{16}$ x 2

Degree 16: $C_4:C_4$, $QD_{16}$ x 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 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,27,26,28)(29,31,30,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 5,12,30)( 2, 6,11,29)( 3, 8,10,31)( 4, 7, 9,32)(13,18,22,27)(14,17,21,28) (15,20,24,25)(16,19,23,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,31, 6,32)( 7,30, 8,29)(13,23,14,24)(15,22,16,21) (17,26,18,25)(19,27,20,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,21)(14,22)(15,23) (16,24)(17,27)(18,28)(19,25)(20,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,22)(14,21)(15,24) (16,23)(17,28)(18,27)(19,26)(20,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,13,11,21)( 2,14,12,22)( 3,16, 9,24)( 4,15,10,23)( 5,19,29,25)( 6,20,30,26) ( 7,18,31,28)( 8,17,32,27)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,17, 3,20, 2,18, 4,19)( 5,24, 7,21, 6,23, 8,22)( 9,26,12,28,10,25,11,27) (13,30,15,32,14,29,16,31)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,18, 3,19, 2,17, 4,20)( 5,23, 7,22, 6,24, 8,21)( 9,25,12,27,10,26,11,28) (13,29,15,31,14,30,16,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,21,11,13)( 2,22,12,14)( 3,24, 9,16)( 4,23,10,15)( 5,25,29,19)( 6,26,30,20) ( 7,28,31,18)( 8,27,32,17)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,25, 4,28, 2,26, 3,27)( 5,14, 8,15, 6,13, 7,16)( 9,17,11,19,10,18,12,20) (21,31,24,29,22,32,23,30)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,26, 4,27, 2,25, 3,28)( 5,13, 8,16, 6,14, 7,15)( 9,18,11,20,10,17,12,19) (21,32,24,30,22,31,23,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,29,12, 6)( 2,30,11, 5)( 3,32,10, 7)( 4,31, 9, 8)(13,28,22,17)(14,27,21,18) (15,26,24,19)(16,25,23,20)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 13]
Character table:   
      2  5  5  4  3  4  5  5  3  4  4  3  4  4  3

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

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

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