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Properties

Label 20T669
Degree $20$
Order $61440$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3^5.D_6$

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Show commands: Magma

magma: G := TransitiveGroup(20, 669);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $669$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^5.D_6$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,15,18,5,4,10,14,19,7)(2,11,16,17,6,3,9,13,20,8), (1,19,8,14,2,20,7,13)(3,17,5,16,4,18,6,15)(9,10)(11,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$120$:  $S_5$
$240$:  $S_5\times C_2$
$1920$:  $(C_2^4:A_5) : C_2$ x 3
$3840$:  $C_2 \wr S_5$ x 3
$30720$:  20T555

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $S_5$

Degree 10: $(C_2^4:A_5) : C_2$, $C_2 \wr S_5$ x 2

Low degree siblings

20T664 x 6, 20T669 x 5, 40T18934 x 6, 40T19059 x 12, 40T19085 x 6, 40T19086 x 6, 40T19087 x 6, 40T19107 x 6, 40T19108 x 6, 40T19114 x 12, 40T19141 x 3, 40T19195 x 6, 40T19196 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 126 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $61440=2^{12} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  61440.b
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);