Properties

Label 15T32
Degree $15$
Order $750$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\wr S_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 32);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\wr S_3$
CHM label:  $[5^{3}]S(3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,6,9,12,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$
$150$:  $(C_5^2 : C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T32 x 3, 30T184 x 4, 30T185 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 65 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $750=2 \cdot 3 \cdot 5^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  750.26
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);