# Properties

 Label 32T29 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 $t$: $29$ Group: $C_4:D_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ Generators: (1,9)(2,10)(3,11)(4,12)(5,32)(6,31)(7,30)(8,29)(13,25)(14,26)(15,28)(16,27)(17,21)(18,22)(19,23)(20,24), (1,14)(2,13)(3,15)(4,16)(5,28)(6,27)(7,25)(8,26)(9,23)(10,24)(11,21)(12,22)(17,32)(18,31)(19,29)(20,30), (1,11,4,10)(2,12,3,9)(5,31,7,29)(6,32,8,30)(13,23,15,22)(14,24,16,21)(17,27,20,26)(18,28,19,25)

## 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 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 12

Degree 8: $C_2^3$, $D_4$ x 6, $D_4\times C_2$ x 12

Degree 16: $D_4\times C_2$ x 3, $C_4^2:C_2$ x 4

## Low degree siblings

16T51 x 4

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 34]
 Character table:  2 5 3 5 4 3 4 4 3 4 3 4 4 5 5 1a 2a 2b 4a 2c 4b 4c 2d 4d 2e 4e 4f 2f 2g 2P 1a 1a 1a 2b 1a 2b 2g 1a 2g 1a 2f 2f 1a 1a 3P 1a 2a 2b 4a 2c 4b 4c 2d 4d 2e 4e 4f 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 . . . . . . . 2 -2 2 -2 X.14 2 . -2 . . . 2 . -2 . . . -2 2