# Properties

 Label 32T2 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $OD_{16}:C_2$

## Group action invariants

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

## 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, $C_2^3$
$16$:  $C_4\times C_2^2$

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, $C_2^3$

Degree 16: $C_4\times C_2^2$, $(C_8:C_2):C_2$ x 3

## Low degree siblings

16T16 x 3

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 38]
 Character table:  2 5 4 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 1a 2a 2b 8a 8b 8c 4a 4b 4c 8d 8e 8f 8g 8h 2c 4d 8i 8j 2d 4e 2P 1a 1a 1a 4c 4c 4c 2b 2b 2b 4b 4b 4b 4b 4c 1a 2b 4b 4c 1a 2b 3P 1a 2a 2b 8d 8f 8e 4a 4c 4b 8a 8c 8b 8j 8i 2c 4d 8h 8g 2d 4e 5P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8d 8f 8e 8g 8h 2c 4d 8i 8j 2d 4e 7P 1a 2a 2b 8d 8e 8f 4a 4c 4b 8a 8b 8c 8j 8i 2c 4d 8h 8g 2d 4e 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 A -A -A 1 -1 -1 -A A A A -A 1 -1 A -A 1 -1 X.10 1 -1 1 -A A A 1 -1 -1 A -A -A -A A 1 -1 -A A 1 -1 X.11 1 -1 1 A -A -A 1 -1 -1 -A A A -A A -1 1 -A A -1 1 X.12 1 -1 1 -A A A 1 -1 -1 A -A -A A -A -1 1 A -A -1 1 X.13 1 1 1 A A A -1 -1 -1 -A -A -A A A -1 -1 -A -A 1 1 X.14 1 1 1 -A -A -A -1 -1 -1 A A A -A -A -1 -1 A A 1 1 X.15 1 1 1 A A A -1 -1 -1 -A -A -A -A -A 1 1 A A -1 -1 X.16 1 1 1 -A -A -A -1 -1 -1 A A A A A 1 1 -A -A -1 -1 X.17 2 . -2 . B -B . C -C . /B -/B . . . . . . . . X.18 2 . -2 . -/B /B . -C C . -B B . . . . . . . . X.19 2 . -2 . /B -/B . -C C . B -B . . . . . . . . X.20 2 . -2 . -B B . C -C . -/B /B . . . . . . . . A = -E(4) = -Sqrt(-1) = -i B = -2*E(8) C = -2*E(4) = -2*Sqrt(-1) = -2i