# Properties

 Label 40T16 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2:C_{20}$

Show commands: Magma

magma: G := TransitiveGroup(40, 16);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$5$:  $C_5$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$10$:  $C_{10}$ x 3
$16$:  $C_2^2:C_4$
$20$:  20T1 x 2, 20T3
$40$:  20T12 x 2, 40T2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 5: $C_5$

Degree 8: $C_2^2:C_4$

Degree 10: $C_{10}$

Degree 20: 20T1, 20T12 x 2

## Low degree siblings

40T16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $80=2^{4} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 80.21 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);