# Properties

 Label 40T247579 Degree $40$ Order $1600000000$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^8.C_2^3.C_2^6.C_2^3$

Show commands: Magma

magma: G := TransitiveGroup(40, 247579);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $247579$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5^8.C_2^3.C_2^6.C_2^3$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,23,40,18,5,24,39,19,4,25,38,20,3,21,37,16,2,22,36,17)(6,26,12,34,10,27,14,31,9,28,11,33,8,29,13,35,7,30,15,32), (1,6,4,10)(2,9,3,7)(5,8)(11,36,13,40)(12,38)(14,37,15,39)(17,20)(18,19)(21,23,24,22)(26,27,30,29)(31,35,34,33,32), (1,26,9,23,5,30,7,22,4,29,10,21,3,28,8,25,2,27,6,24)(11,17,39,31,15,16,40,34,14,20,36,32,13,19,37,35,12,18,38,33) magma: Generators(G);

## 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, $C_2^3 : C_4$ x 4
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 5: None

Degree 8: $C_2^2 \wr C_2$

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $1600000000=2^{12} \cdot 5^{8}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 1600000000.iyu magma: IdentifyGroup(G); Character table: not computed

magma: CharacterTable(G);