# Properties

 Label 32T31 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $D_{16}$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
16:  $D_{8}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $D_{8}$ x 2

Degree 16: $D_{8}$, $D_{16}$ x 2

## Low degree siblings

16T56 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 2)( 3, 4)( 5,29)( 6,30)( 7,31)( 8,32)( 9,28)(10,27)(11,26)(12,25)(13,24) (14,23)(15,22)(16,21)(17,20)(18,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$ $16, 16$ $2$ $16$ $( 1, 5, 9,15,17,24,26,31, 3, 8,12,14,19,21,27,30)( 2, 6,10,16,18,23,25,32, 4, 7,11,13,20,22,28,29)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28) (16,27)(17,22)(18,21)(19,23)(20,24)$ $16, 16$ $2$ $16$ $( 1, 8, 9,14,17,21,26,30, 3, 5,12,15,19,24,27,31)( 2, 7,10,13,18,22,25,29, 4, 6,11,16,20,23,28,32)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 9,17,26, 3,12,19,27)( 2,10,18,25, 4,11,20,28)( 5,15,24,31, 8,14,21,30) ( 6,16,23,32, 7,13,22,29)$ $8, 8, 8, 8$ $2$ $8$ $( 1,12,17,27, 3, 9,19,26)( 2,11,18,28, 4,10,20,25)( 5,14,24,30, 8,15,21,31) ( 6,13,23,29, 7,16,22,32)$ $16, 16$ $2$ $16$ $( 1,14,26, 5,19,31, 9,21, 3,15,27, 8,17,30,12,24)( 2,13,25, 6,20,32,10,22, 4, 16,28, 7,18,29,11,23)$ $16, 16$ $2$ $16$ $( 1,15,26, 8,19,30, 9,24, 3,14,27, 5,17,31,12,21)( 2,16,25, 7,20,29,10,23, 4, 13,28, 6,18,32,11,22)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,17, 3,19)( 2,18, 4,20)( 5,24, 8,21)( 6,23, 7,22)( 9,26,12,27)(10,25,11,28) (13,29,16,32)(14,30,15,31)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 18]
 Character table:  2 5 2 5 4 2 4 4 4 4 4 4 1a 2a 2b 16a 2c 16b 8a 8b 16c 16d 4a 2P 1a 1a 1a 8a 1a 8a 4a 4a 8b 8b 2b 3P 1a 2a 2b 16d 2c 16c 8b 8a 16a 16b 4a 5P 1a 2a 2b 16c 2c 16d 8b 8a 16b 16a 4a 7P 1a 2a 2b 16b 2c 16a 8a 8b 16d 16c 4a 11P 1a 2a 2b 16c 2c 16d 8b 8a 16b 16a 4a 13P 1a 2a 2b 16d 2c 16c 8b 8a 16a 16b 4a X.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 X.3 1 -1 1 1 -1 1 1 1 1 1 1 X.4 1 1 1 -1 -1 -1 1 1 -1 -1 1 X.5 2 . 2 . . . -2 -2 . . 2 X.6 2 . 2 A . A . . -A -A -2 X.7 2 . 2 -A . -A . . A A -2 X.8 2 . -2 B . -B -A A -C C . X.9 2 . -2 C . -C A -A B -B . X.10 2 . -2 -C . C A -A -B B . X.11 2 . -2 -B . B -A A C -C . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 B = -E(16)+E(16)^7 C = -E(16)^3+E(16)^5