# Properties

 Label 40T59587 Degree $40$ Order $163840$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^6.C_2^8:C_{10}$

Show commands: Magma

magma: G := TransitiveGroup(40, 59587);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $59587$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2^6.C_2^8:C_{10}$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,9,18,39,32)(2,10,17,40,31)(3,12,19,37,30)(4,11,20,38,29)(5,13,21,36,26)(6,14,22,35,25)(7,16,23,34,28)(8,15,24,33,27), (1,15,17,33,27,2,16,18,34,28)(3,14,19,35,25)(4,13,20,36,26)(5,11,21,38,30,6,12,22,37,29)(7,10,24,40,32)(8,9,23,39,31), (1,19,32,12,37,7,21,28,14,36)(2,20,31,11,38,8,22,27,13,35)(3,18,30,10,39,6,24,25,15,33)(4,17,29,9,40,5,23,26,16,34) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3
$80$:  $C_2^4 : C_5$ x 17
$160$:  $C_2 \times (C_2^4 : C_5)$ x 51

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 8: None

Degree 10: $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$ x 2

Degree 20: 20T263

## 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

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/4$

## Group invariants

 Order: $163840=2^{15} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 163840.hih magma: IdentifyGroup(G); Character table: not computed

magma: CharacterTable(G);