# Properties

 Label 32T16 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4.D_4$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $16$ Group: $C_4.D_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ Generators: (1,22,31,25)(2,21,32,26)(3,23,30,28)(4,24,29,27)(5,17,10,14)(6,18,9,13)(7,19,12,16)(8,20,11,15), (1,8)(2,7)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,16,30,18)(2,15,29,17)(3,13,31,19)(4,14,32,20)(5,26,11,24)(6,25,12,23)(7,28,9,22)(8,27,10,21)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4, $Q_8:C_2$ x 6

Degree 16: $D_4\times C_2$, $Q_8 : C_2$ x 2, 16T30 x 2

## Low degree siblings

16T30 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 31]
 Character table:  2 5 3 5 3 4 4 4 3 4 3 4 4 5 5 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 2d 2e 2P 1a 1a 1a 1a 2b 2b 2d 2b 2d 2b 2e 2e 1a 1a 3P 1a 2a 2b 2c 4a 4b 4e 4d 4c 4f 4h 4g 2d 2e 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 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 X.7 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 X.9 2 . 2 . 2 -2 . . . . . . -2 -2 X.10 2 . 2 . -2 2 . . . . . . -2 -2 X.11 2 . -2 . . . . . . . A -A 2 -2 X.12 2 . -2 . . . . . . . -A A 2 -2 X.13 2 . -2 . . . A . -A . . . -2 2 X.14 2 . -2 . . . -A . A . . . -2 2 A = -2*E(4) = -2*Sqrt(-1) = -2i