# Properties

 Label 32T1 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2\times OD_{16}$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_8:C_2$ x 2, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $C_8:C_2$ x 2

Degree 16: $C_4\times C_2^2$, $C_8: C_2$ x 2, $C_2 \times (C_8:C_2)$ x 2

## Low degree siblings

16T15 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 2)( 3, 4)( 5,22)( 6,21)( 7,23)( 8,24)( 9,10)(11,12)(13,29)(14,30)(15,32) (16,31)(17,18)(19,20)(25,26)(27,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 4)( 2, 3)( 5,24)( 6,23)( 7,21)( 8,22)( 9,12)(10,11)(13,32)(14,31)(15,29) (16,30)(17,19)(18,20)(25,27)(26,28)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 5, 9,13,20,21,27,30)( 2, 6,10,14,19,22,28,29)( 3, 8,11,15,17,23,26,31) ( 4, 7,12,16,18,24,25,32)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 6,27,29,20,22, 9,14)( 2, 5,28,30,19,21,10,13)( 3, 7,26,32,17,24,11,16) ( 4, 8,25,31,18,23,12,15)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 7,27,32,20,24, 9,16)( 2, 8,28,31,19,23,10,15)( 3, 6,26,29,17,22,11,14) ( 4, 5,25,30,18,21,12,13)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 8, 9,15,20,23,27,31)( 2, 7,10,16,19,24,28,32)( 3, 5,11,13,17,21,26,30) ( 4, 6,12,14,18,22,25,29)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1, 9,20,27)( 2,10,19,28)( 3,11,17,26)( 4,12,18,25)( 5,13,21,30)( 6,14,22,29) ( 7,16,24,32)( 8,15,23,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,10,20,28)( 2, 9,19,27)( 3,12,17,25)( 4,11,18,26)( 5,29,21,14)( 6,30,22,13) ( 7,31,24,15)( 8,32,23,16)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1,11,20,26)( 2,12,19,25)( 3, 9,17,27)( 4,10,18,28)( 5,15,21,31)( 6,16,22,32) ( 7,14,24,29)( 8,13,23,30)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,12,20,25)( 2,11,19,26)( 3,10,17,28)( 4, 9,18,27)( 5,32,21,16)( 6,31,22,15) ( 7,30,24,13)( 8,29,23,14)$ $8, 8, 8, 8$ $2$ $8$ $( 1,13,27, 5,20,30, 9,21)( 2,14,28, 6,19,29,10,22)( 3,15,26, 8,17,31,11,23) ( 4,16,25, 7,18,32,12,24)$ $8, 8, 8, 8$ $2$ $8$ $( 1,14, 9,22,20,29,27, 6)( 2,13,10,21,19,30,28, 5)( 3,16,11,24,17,32,26, 7) ( 4,15,12,23,18,31,25, 8)$ $8, 8, 8, 8$ $2$ $8$ $( 1,15,27, 8,20,31, 9,23)( 2,16,28, 7,19,32,10,24)( 3,13,26, 5,17,30,11,21) ( 4,14,25, 6,18,29,12,22)$ $8, 8, 8, 8$ $2$ $8$ $( 1,16, 9,24,20,32,27, 7)( 2,15,10,23,19,31,28, 8)( 3,14,11,22,17,29,26, 6) ( 4,13,12,21,18,30,25, 5)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,23)( 6,24)( 7,22)( 8,21)( 9,26)(10,25)(11,27) (12,28)(13,31)(14,32)(15,30)(16,29)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,24)( 8,23)( 9,27)(10,28)(11,26) (12,25)(13,30)(14,29)(15,31)(16,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1,26,20,11)( 2,25,19,12)( 3,27,17, 9)( 4,28,18,10)( 5,31,21,15)( 6,32,22,16) ( 7,29,24,14)( 8,30,23,13)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1,27,20, 9)( 2,28,19,10)( 3,26,17,11)( 4,25,18,12)( 5,30,21,13)( 6,29,22,14) ( 7,32,24,16)( 8,31,23,15)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 37]
 Character table:  2 5 4 5 4 4 4 4 4 5 4 5 4 4 4 4 4 5 5 5 5 1a 2a 2b 2c 8a 8b 8c 8d 4a 4b 4c 4d 8e 8f 8g 8h 2d 2e 4e 4f 2P 1a 1a 1a 1a 4a 4f 4f 4a 2e 2e 2e 2e 4f 4a 4f 4a 1a 1a 2e 2e 3P 1a 2a 2b 2c 8e 8f 8h 8g 4f 4b 4e 4d 8a 8b 8d 8c 2d 2e 4c 4a 5P 1a 2a 2b 2c 8a 8b 8c 8d 4a 4b 4c 4d 8e 8f 8g 8h 2d 2e 4e 4f 7P 1a 2a 2b 2c 8e 8f 8h 8g 4f 4b 4e 4d 8a 8b 8d 8c 2d 2e 4c 4a X.1 1 1 1 1 1 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 1 -1 -1 1 -1 1 X.3 1 -1 -1 1 1 -1 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 -1 1 1 1 1 1 X.5 1 -1 1 -1 1 -1 -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 1 1 -1 1 -1 1 X.7 1 1 -1 -1 1 1 -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 -1 -1 1 1 1 1 X.9 1 -1 -1 1 A -A A -A -1 1 1 -1 -A A A -A -1 1 1 -1 X.10 1 -1 -1 1 -A A -A A -1 1 1 -1 A -A -A A -1 1 1 -1 X.11 1 -1 1 -1 A -A -A A -1 1 -1 1 -A A -A A 1 1 -1 -1 X.12 1 -1 1 -1 -A A A -A -1 1 -1 1 A -A A -A 1 1 -1 -1 X.13 1 1 -1 -1 A A -A -A -1 -1 1 1 -A -A A A -1 1 1 -1 X.14 1 1 -1 -1 -A -A A A -1 -1 1 1 A A -A -A -1 1 1 -1 X.15 1 1 1 1 A A A A -1 -1 -1 -1 -A -A -A -A 1 1 -1 -1 X.16 1 1 1 1 -A -A -A -A -1 -1 -1 -1 A A A A 1 1 -1 -1 X.17 2 . -2 . . . . . B . -B . . . . . 2 -2 B -B X.18 2 . -2 . . . . . -B . B . . . . . 2 -2 -B B X.19 2 . 2 . . . . . B . B . . . . . -2 -2 -B -B X.20 2 . 2 . . . . . -B . -B . . . . . -2 -2 B B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i