# Properties

 Label 40T19 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times D_4:C_2$

# Learn more about

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $19$ Group: $C_5\times D_4:C_2$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $20$ Generators: (3,4)(7,8)(11,12)(13,14)(19,20)(23,24)(25,26)(29,30)(33,34)(39,40), (1,30,18,7,35,23,10,39,28,13)(2,29,17,8,36,24,9,40,27,14)(3,31,19,5,34,21,11,37,25,15)(4,32,20,6,33,22,12,38,26,16), (1,34,28,19,10,3,35,25,18,11)(2,33,27,20,9,4,36,26,17,12)(5,40,31,24,15,8,37,29,21,14)(6,39,32,23,16,7,38,30,22,13)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$5$:  $C_5$
$8$:  $C_2^3$
$10$:  $C_{10}$ x 7
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 8: $Q_8:C_2$

Degree 10: $C_{10}$ x 3

Degree 20: 20T3

## Low degree siblings

There are no siblings with degree $\leq 10$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

There are 50 conjugacy classes of elements. Data not shown.

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 48]
 Character table: not available.