# Properties

 Label 32T18 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $SD_{16}:C_2$

## Group action invariants

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

## 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, $C_2^3$
$16$:  $D_4\times C_2$

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

Degree 16: $D_4\times C_2$, 16T32, 16T50

## Low degree siblings

16T32, 16T50

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 44]
 Character table:  2 5 3 5 4 3 4 3 3 3 3 4 1a 2a 2b 4a 4b 2c 8a 4c 8b 4d 4e 2P 1a 1a 1a 2b 2b 1a 4e 2b 4e 2b 2b 3P 1a 2a 2b 4a 4b 2c 8a 4c 8b 4d 4e 5P 1a 2a 2b 4a 4b 2c 8a 4c 8b 4d 4e 7P 1a 2a 2b 4a 4b 2c 8a 4c 8b 4d 4e X.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 X.3 1 -1 1 -1 1 -1 1 -1 -1 1 1 X.4 1 -1 1 1 -1 1 -1 1 -1 1 1 X.5 1 -1 1 1 -1 1 1 -1 1 -1 1 X.6 1 1 1 -1 -1 -1 -1 -1 1 1 1 X.7 1 1 1 -1 -1 -1 1 1 -1 -1 1 X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 X.9 2 . 2 -2 . 2 . . . . -2 X.10 2 . 2 2 . -2 . . . . -2 X.11 4 . -4 . . . . . . . .