Properties

Label 32T44
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4:C_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $44$
Group :  $C_4:C_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,5,9,13,17,23,25,29)(2,6,10,14,18,24,26,30)(3,8,11,16,19,22,27,32)(4,7,12,15,20,21,28,31), (1,22,9,32,17,8,25,16)(2,21,10,31,18,7,26,15)(3,24,11,30,19,6,27,14)(4,23,12,29,20,5,28,13)
$|\Aut(F/K)|$:  $32$

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:  $D_{4}$, $C_8$ x 2, $C_4\times C_2$, $Q_8$
16:  $C_8:C_2$, $C_8\times C_2$, $C_4:C_4$

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: $C_8$ x 2, $C_4\times C_2$, $D_4$, $Q_8$, $C_8:C_2$

Degree 16: $C_8\times C_2$, $C_8: C_2$, $C_4:C_4$

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 12]
Character table:   
      2  5  5  4   4   4  5  5  4   4   4  5  5  4   4   4  5  5  4   4   4

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

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   A  -A -1 -1  1  -A   A  1  1 -1  -A   A -1 -1  1  -A   A
X.6      1  1 -1  -A   A -1 -1  1   A  -A  1  1 -1   A  -A -1 -1  1   A  -A
X.7      1  1 -1   B  -B -A -A  A -/B  /B -1 -1  1   B  -B  A  A -A  /B -/B
X.8      1  1 -1 -/B  /B  A  A -A   B  -B -1 -1  1 -/B  /B -A -A  A  -B   B
X.9      1  1 -1  /B -/B  A  A -A  -B   B -1 -1  1  /B -/B -A -A  A   B  -B
X.10     1  1 -1  -B   B -A -A  A  /B -/B -1 -1  1  -B   B  A  A -A -/B  /B
X.11     1  1  1   A   A -1 -1 -1  -A  -A  1  1  1   A   A -1 -1 -1  -A  -A
X.12     1  1  1  -A  -A -1 -1 -1   A   A  1  1  1  -A  -A -1 -1 -1   A   A
X.13     1  1  1   B   B -A -A -A -/B -/B -1 -1 -1  -B  -B  A  A  A  /B  /B
X.14     1  1  1 -/B -/B  A  A  A   B   B -1 -1 -1  /B  /B -A -A -A  -B  -B
X.15     1  1  1  /B  /B  A  A  A  -B  -B -1 -1 -1 -/B -/B -A -A -A   B   B
X.16     1  1  1  -B  -B -A -A -A  /B  /B -1 -1 -1   B   B  A  A  A -/B -/B
X.17     2 -2  .   .   . -2  2  .   .   .  2 -2  .   .   . -2  2  .   .   .
X.18     2 -2  .   .   .  2 -2  .   .   .  2 -2  .   .   .  2 -2  .   .   .
X.19     2 -2  .   .   .  C -C  .   .   . -2  2  .   .   . -C  C  .   .   .
X.20     2 -2  .   .   . -C  C  .   .   . -2  2  .   .   .  C -C  .   .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)
C = -2*E(4)
  = -2*Sqrt(-1) = -2i