# Properties

 Label 32T15 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2\times D_8$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $15$ Group: $C_2\times D_8$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $32$ Generators: (1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,10)(11,12)(21,30)(22,29)(23,31)(24,32)(25,26)(27,28), (1,24)(2,23)(3,21)(4,22)(5,17)(6,18)(7,20)(8,19)(9,14)(10,13)(11,15)(12,16)(25,30)(26,29)(27,32)(28,31), (1,11,20,26)(2,12,19,25)(3,9,17,28)(4,10,18,27)(5,14,21,31)(6,13,22,32)(7,15,24,29)(8,16,23,30)

## 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, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$

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, $D_{8}$ x 4, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, $D_{8}$ x 2, 16T29 x 4

## Low degree siblings

16T29 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 4)( 5,30)( 6,29)( 7,32)( 8,31)( 9,27)(10,28)(11,25)(12,26)(13,24) (14,23)(15,22)(16,21)(17,18)(19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 4)( 2, 3)( 5,32)( 6,31)( 7,30)( 8,29)( 9,25)(10,26)(11,27)(12,28)(13,21) (14,22)(15,23)(16,24)(17,19)(18,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,28) (16,27)(17,24)(18,23)(19,22)(20,21)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 6, 9,16,20,22,28,30)( 2, 5,10,15,19,21,27,29)( 3, 8,11,13,17,23,26,32) ( 4, 7,12,14,18,24,25,31)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,26) (16,25)(17,21)(18,22)(19,23)(20,24)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 8, 9,13,20,23,28,32)( 2, 7,10,14,19,24,27,31)( 3, 6,11,16,17,22,26,30) ( 4, 5,12,15,18,21,25,29)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 9,20,28)( 2,10,19,27)( 3,11,17,26)( 4,12,18,25)( 5,15,21,29)( 6,16,22,30) ( 7,14,24,31)( 8,13,23,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,11,20,26)( 2,12,19,25)( 3, 9,17,28)( 4,10,18,27)( 5,14,21,31)( 6,13,22,32) ( 7,15,24,29)( 8,16,23,30)$ $8, 8, 8, 8$ $2$ $8$ $( 1,13,28, 8,20,32, 9,23)( 2,14,27, 7,19,31,10,24)( 3,16,26, 6,17,30,11,22) ( 4,15,25, 5,18,29,12,21)$ $8, 8, 8, 8$ $2$ $8$ $( 1,16,28, 6,20,30, 9,22)( 2,15,27, 5,19,29,10,21)( 3,13,26, 8,17,32,11,23) ( 4,14,25, 7,18,31,12,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,21)( 8,22)( 9,26)(10,25)(11,28) (12,27)(13,30)(14,29)(15,31)(16,32)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,24)( 8,23)( 9,28)(10,27)(11,26) (12,25)(13,32)(14,31)(15,29)(16,30)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 39]
 Character table:  2 5 3 5 3 3 4 3 4 4 4 4 4 5 5 1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 2f 2g 2P 1a 1a 1a 1a 1a 4a 1a 4a 2g 2g 4a 4a 1a 1a 3P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g 5P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g 7P 1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 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 . . A . -A . . A -A 2 -2 X.12 2 . -2 . . -A . A . . -A A 2 -2 X.13 2 . 2 . . A . A . . -A -A -2 -2 X.14 2 . 2 . . -A . -A . . A A -2 -2 A = -E(8)+E(8)^3 = -Sqrt(2) = -r2