# Properties

 Label 32T49 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2\times Q_{16}$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $49$ Group: $C_2\times Q_{16}$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $32$ Generators: (1,30,2,29)(3,32,4,31)(5,10,6,9)(7,11,8,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28), (1,11,2,12)(3,9,4,10)(5,31,6,32)(7,30,8,29)(13,28,14,27)(15,25,16,26)(17,24,18,23)(19,21,20,22), (1,23,2,24)(3,21,4,22)(5,17,6,18)(7,20,8,19)(9,13,10,14)(11,16,12,15)(25,32,26,31)(27,29,28,30)

## 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, $C_2^3$
$16$:  $D_4\times C_2$, $Q_{16}$ 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

Degree 16: $D_4\times C_2$, $Q_{16}$ x 2

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 41]
 Character table:  2 5 5 3 3 5 5 4 3 4 4 4 4 3 4 1a 2a 4a 4b 2b 2c 4c 4d 8a 8b 8c 8d 4e 4f 2P 1a 1a 2a 2a 1a 1a 2a 2a 4f 4f 4f 4f 2a 2a 3P 1a 2a 4a 4b 2b 2c 4c 4d 8b 8a 8d 8c 4e 4f 5P 1a 2a 4a 4b 2b 2c 4c 4d 8b 8a 8d 8c 4e 4f 7P 1a 2a 4a 4b 2b 2c 4c 4d 8a 8b 8c 8d 4e 4f 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 . . A -A -A A . . X.12 2 -2 . . -2 2 . . -A A A -A . . X.13 2 -2 . . 2 -2 . . A -A A -A . . X.14 2 -2 . . 2 -2 . . -A A -A A . . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2