↑

Properties

Label 15T39
Degree $15$
Order $1500$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^3:A_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(15, 39);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^3:A_4$
CHM label:  $[1/2.D(5)^{3}]3$
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)(7,13)(11,14), (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)
$3$:  $C_3$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

20T209, 30T271 x 2, 30T277, 30T279, 30T281

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 $ $1$ $1$ $()$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 6, 9,12,15)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 8,14, 5,11)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $75$ $2$ $( 4,13)( 5,14)( 7,10)( 8,11)$
$ 5, 2, 2, 2, 2, 1, 1 $ $150$ $10$ $( 3, 6, 9,12,15)( 4,13)( 5,14)( 7,10)( 8,11)$
$ 5, 2, 2, 2, 2, 1, 1 $ $150$ $10$ $( 3, 9,15, 6,12)( 4,13)( 5,14)( 7,10)( 8,11)$
$ 3, 3, 3, 3, 3 $ $100$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 15 $ $100$ $15$ $( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)$
$ 15 $ $100$ $15$ $( 1,12, 2, 7, 3, 8,13, 9,14, 4,15, 5,10, 6,11)$
$ 15 $ $100$ $15$ $( 1, 3, 8,13,15, 5,10,12, 2, 7, 9,14, 4, 6,11)$
$ 15 $ $100$ $15$ $( 1,15, 5,10, 9,14, 4, 3, 8,13,12, 2, 7, 6,11)$
$ 3, 3, 3, 3, 3 $ $100$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $100$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 15 $ $100$ $15$ $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$
$ 15 $ $100$ $15$ $( 1,11, 3,13, 8,15,10, 5,12, 7, 2, 9, 4,14, 6)$
$ 15 $ $100$ $15$ $( 1,11,15,10, 5, 9, 4,14, 3,13, 8,12, 7, 2, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1500=2^{2} \cdot 3 \cdot 5^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1500.123
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);