# Properties

 Label 32T20 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^2:D_4$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $20$ Group: $C_2^2:D_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ Generators: (1,10)(2,9)(3,12)(4,11)(5,32)(6,31)(7,29)(8,30)(13,22)(14,21)(15,24)(16,23)(17,25)(18,26)(19,27)(20,28), (1,25,4,28)(2,26,3,27)(5,16,8,13)(6,15,7,14)(9,19,12,18)(10,20,11,17)(21,29,24,31)(22,30,23,32), (1,21,32,26)(2,22,31,25)(3,23,29,28)(4,24,30,27)(5,18,10,14)(6,17,9,13)(7,20,12,16)(8,19,11,15)

## 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 4, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 8

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

Degree 16: $D_4\times C_2$ x 2, $Q_8 : C_2$, 16T34 x 2, 16T43 x 2

## Low degree siblings

16T34 x 2, 16T43 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,28) (22,27)(23,26)(24,25)(29,30)(31,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,31)(30,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,31)(10,32)(11,30)(12,29)(13,25)(14,26)(15,27) (16,28)(17,22)(18,21)(19,24)(20,23)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1, 6, 4, 7)( 2, 5, 3, 8)( 9,30,12,32)(10,29,11,31)(13,21,16,24)(14,22,15,23) (17,26,20,27)(18,25,19,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,32)( 6,31)( 7,29)( 8,30)(13,22)(14,21)(15,24) (16,23)(17,25)(18,26)(19,27)(20,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22) (12,21)(17,32)(18,31)(19,29)(20,30)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,14,32,18)( 2,13,31,17)( 3,16,29,20)( 4,15,30,19)( 5,26,10,21)( 6,25, 9,22) ( 7,28,12,23)( 8,27,11,24)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,15,32,19)( 2,16,31,20)( 3,13,29,17)( 4,14,30,18)( 5,27,10,24)( 6,28, 9,23) ( 7,25,12,22)( 8,26,11,21)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,21,32,26)( 2,22,31,25)( 3,23,29,28)( 4,24,30,27)( 5,18,10,14)( 6,17, 9,13) ( 7,20,12,16)( 8,19,11,15)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,22, 4,23)( 2,21, 3,24)( 5,20, 8,17)( 6,19, 7,18)( 9,15,12,14)(10,16,11,13) (25,30,28,32)(26,29,27,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,24,32,27)( 2,23,31,28)( 3,22,29,25)( 4,21,30,26)( 5,19,10,15)( 6,20, 9,16) ( 7,17,12,13)( 8,18,11,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,20)(14,19)(15,18) (16,17)(21,27)(22,28)(23,25)(24,26)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,32)( 2,31)( 3,29)( 4,30)( 5,10)( 6, 9)( 7,12)( 8,11)(13,17)(14,18)(15,19) (16,20)(21,26)(22,25)(23,28)(24,27)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 28]
 Character table:  2 5 3 5 4 3 4 3 4 4 4 3 4 5 5 1a 2a 2b 2c 4a 2d 2e 4b 4c 4d 4e 4f 2f 2g 2P 1a 1a 1a 1a 2b 1a 1a 2g 2g 2g 2b 2g 1a 1a 3P 1a 2a 2b 2c 4a 2d 2e 4b 4c 4f 4e 4d 2f 2g 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 . . . . -2 2 . . . -2 2 X.12 2 . -2 . . . . 2 -2 . . . -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