# Properties

 Label 40T224732 Degree $40$ Order $400000000$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $224732$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,15,5,12,3,11,4,14)(2,13)(6,39)(7,36,8,38,10,37,9,40)(16,26,18,29,17,30,20,27)(19,28)(21,33,25,31,23,32,24,34)(22,35), (1,32)(2,34)(3,31)(4,33)(5,35)(6,17,27,39,10,19,30,40,9,16,28,36,8,18,26,37,7,20,29,38)(11,24,15,21)(12,22,14,23)(13,25), (1,7,5,8)(2,6,4,9)(3,10)(11,16,15,19,14,17,13,20,12,18)(21,30,25,28)(22,27,24,26)(23,29)(31,39,35,38,34,37,33,36,32,40)

## 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
$8$:  $D_{4}$ x 14, $C_4\times C_2$ x 6, $C_2^3$, $Q_8$ x 2
$16$:  $D_4\times C_2$ x 7, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 6
$32$:  $C_2^2 \wr C_2$ x 3
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Degree 10: None

Degree 20: None

## Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

## Conjugacy classes

There are 820 conjugacy classes of elements. Data not shown.

## Group invariants

 Order: $400000000=2^{10} \cdot 5^{8}$ Cyclic: no Abelian: no Solvable: yes GAP id: not available
 Character table: not available.