# Properties

 Label 40T20 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_4\times C_{10}$

Show commands: Magma

magma: G := TransitiveGroup(40, 20);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $20$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_4\times C_{10}$ 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: (1,34,27,20,10,3,36,26,18,11,2,33,28,19,9,4,35,25,17,12)(5,39,32,24,15,7,38,29,21,13,6,40,31,23,16,8,37,30,22,14), (1,21)(2,22)(3,24)(4,23)(5,28)(6,27)(7,26)(8,25)(9,32)(10,31)(11,29)(12,30)(13,33)(14,34)(15,35)(16,36)(17,38)(18,37)(19,40)(20,39), (1,35,28,18,10)(2,36,27,17,9)(3,33,25,20,11,4,34,26,19,12)(5,37,31,21,15)(6,38,32,22,16)(7,40,30,24,13,8,39,29,23,14) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$5$:  $C_5$
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $C_{10}$ x 7
$16$:  $D_4\times C_2$
$20$:  20T3 x 7
$40$:  20T12 x 2, 40T7

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $C_5$

Degree 8: $D_4\times C_2$

Degree 10: $C_{10}$ x 3

Degree 20: 20T3, 20T12 x 2

## Low degree siblings

40T20 x 3

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: 80.46 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);