# Properties

 Label 32T38 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4\times Q_8$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $38$ Group: $C_4\times Q_8$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ 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)

## 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
$16$:  $Q_8:C_2$, $C_4\times C_2^2$, $Q_8\times C_2$

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

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

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

## 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