# Properties

 Label 16T36 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^3.C_4$

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## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $36$ Group: $C_2^3.C_4$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $8$ Generators: (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(15,16)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_4\times C_2$, $(C_8:C_2):C_2$ x 2

## Low degree siblings

8T16 x 2, 16T41 x 2, 32T22

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 3,12)( 4,11)( 7,16)( 8,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$ $8, 8$ $4$ $8$ $( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$ $8, 8$ $4$ $8$ $( 1, 4, 6,16,10,11,13, 7)( 2, 3, 5,15, 9,12,14, 8)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,12,15)( 4, 7,11,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,12, 8)( 4,16,11, 7)$ $8, 8$ $4$ $8$ $( 1, 7,13,11,10,16, 6, 4)( 2, 8,14,12, 9,15, 5, 3)$ $8, 8$ $4$ $8$ $( 1, 8, 6,12,10,15,13, 3)( 2, 7, 5,11, 9,16,14, 4)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 7]
 Character table:  2 5 4 3 3 3 3 4 4 3 3 5 1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d 2P 1a 1a 1a 4a 4b 1a 2d 2d 4b 4a 1a 3P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d 5P 1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d 7P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d 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 1 1 -1 A -A 1 -1 -1 A -A 1 X.6 1 1 -1 -A A 1 -1 -1 -A A 1 X.7 1 1 1 A A -1 -1 -1 -A -A 1 X.8 1 1 1 -A -A -1 -1 -1 A A 1 X.9 2 -2 . . . . -2 2 . . 2 X.10 2 -2 . . . . 2 -2 . . 2 X.11 4 . . . . . . . . . -4 A = -E(4) = -Sqrt(-1) = -i