# Properties

 Label 32T17 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^2:Q_8$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $17$ Group: $C_2^2:Q_8$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ Generators: (1,12,4,10)(2,11,3,9)(5,31,7,30)(6,32,8,29)(13,28,15,26)(14,27,16,25)(17,23,19,22)(18,24,20,21), (1,24,4,21)(2,23,3,22)(5,17,7,19)(6,18,8,20)(9,14,11,16)(10,13,12,15)(25,31,27,30)(26,32,28,29), (1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,28)(24,27)(29,31)(30,32)

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

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

Degree 16: $Q_8\times C_2$, $D_4\times C_2$, $Q_8 : C_2$, 16T31 x 2

## Low degree siblings

16T31 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,19)(14,20)(15,17)(16,18)(21,27) (22,28)(23,26)(24,25)(29,30)(31,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20)(21,25) (22,26)(23,28)(24,27)(29,31)(30,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,29,11,32)(10,30,12,31)(13,27,15,25)(14,28,16,26) (17,24,19,21)(18,23,20,22)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,30,11,31)(10,29,12,32)(13,21,15,24)(14,22,16,23) (17,25,19,27)(18,26,20,28)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,31,11,30)(10,32,12,29)(13,24,15,21)(14,23,16,22) (17,27,19,25)(18,28,20,26)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,29, 7,32)( 6,30, 8,31)(13,23,15,22)(14,24,16,21) (17,28,19,26)(18,27,20,25)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,13,32,18)( 2,14,31,17)( 3,16,30,19)( 4,15,29,20)( 5,25, 9,22)( 6,26,10,21) ( 7,27,11,23)( 8,28,12,24)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,14, 4,16)( 2,13, 3,15)( 5,26, 7,28)( 6,25, 8,27)( 9,21,11,24)(10,22,12,23) (17,29,19,32)(18,30,20,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,21, 4,24)( 2,22, 3,23)( 5,19, 7,17)( 6,20, 8,18)( 9,16,11,14)(10,15,12,13) (25,30,27,31)(26,29,28,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,22,32,25)( 2,21,31,26)( 3,24,30,28)( 4,23,29,27)( 5,20, 9,15)( 6,19,10,16) ( 7,18,11,13)( 8,17,12,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,20)(14,19)(15,18) (16,17)(21,28)(22,27)(23,25)(24,26)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5, 9)( 6,10)( 7,11)( 8,12)(13,18)(14,17)(15,20) (16,19)(21,26)(22,25)(23,27)(24,28)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 29]
 Character table:  2 5 4 4 5 4 4 4 4 3 3 3 3 5 5 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 2d 2e 2P 1a 1a 1a 1a 2c 2c 2c 2c 2e 2c 2c 2e 1a 1a 3P 1a 2a 2b 2c 4a 4c 4b 4d 4e 4f 4g 4h 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 -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