↑

Properties

Label 15T43
Degree $15$
Order $1620$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:D_{10}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(15, 43);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:D_{10}$
CHM label:  $[3^{4}:2]D(5)$
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:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(4,14,9), (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T43 x 7, 30T290 x 8, 30T291 x 8, 30T297 x 8, 45T205 x 16, 45T206 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 5, 5, 5 $ $162$ $5$ $( 1,12, 3, 4,10)( 2, 8, 9,15, 6)( 5,11, 7,13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $90$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $81$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 10, 5 $ $162$ $10$ $( 1, 4,12, 5,13, 6,14, 2,15, 3)( 7,10, 8,11, 9)$
$ 10, 5 $ $162$ $10$ $( 1,12,13, 9, 5,11, 2, 8,14,15)( 3, 4,10, 6, 7)$
$ 6, 2, 2, 2, 2, 1 $ $90$ $6$ $( 2,15,12, 5, 7,10)( 3, 4)( 6,11)( 8,14)( 9,13)$
$ 6, 6, 2, 1 $ $90$ $6$ $( 1,11)( 2,15,12, 5, 7,10)( 3, 9,13,14, 8, 4)$
$ 6, 6, 2, 1 $ $90$ $6$ $( 1, 6)( 2,15,12, 5, 7,10)( 3,14, 8, 9,13, 4)$
$ 6, 2, 2, 2, 2, 1 $ $90$ $6$ $( 2,15)( 3,14, 8, 9,13, 4)( 5,12)( 6,11)( 7,10)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 1,11)( 2,15)( 3, 4)( 5,12)( 7,10)( 8,14)( 9,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1620=2^{2} \cdot 3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1620.422
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);