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Properties

Label 15T70
Degree $15$
Order $19440$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3^4:(C_2\times S_5)$

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Show commands: Magma

magma: G := TransitiveGroup(15, 70);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $70$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:(C_2\times S_5)$
CHM label:  $[3^{4}:2]S(5)$
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:  (1,6,11)(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), (1,4)(6,9)(11,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$120$:  $S_5$
$240$:  $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $S_5$

Low degree siblings

30T910, 30T913, 30T915, 30T917, 30T918, 30T919, 45T615, 45T616

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 $ $20$ $3$ $( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $30$ $3$ $( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $81$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $( 1, 4)( 6, 9)(11,14)$
$ 6, 3, 1, 1, 1, 1, 1, 1 $ $180$ $6$ $( 1,14,11, 9, 6, 4)( 5,10,15)$
$ 3, 3, 2, 2, 2, 1, 1, 1 $ $180$ $6$ $( 1, 4)( 2,12, 7)( 3, 8,13)( 6, 9)(11,14)$
$ 6, 3, 3, 3 $ $180$ $6$ $( 1,14,11, 9, 6, 4)( 2,12, 7)( 3, 8,13)( 5,10,15)$
$ 6, 3, 3, 1, 1, 1 $ $180$ $6$ $( 1,14,11, 9, 6, 4)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 2, 2, 2 $ $60$ $6$ $( 1, 4)( 2, 7,12)( 3, 8,13)( 5,10,15)( 6, 9)(11,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $270$ $2$ $( 1, 4)( 6,14)( 7,12)( 8,13)( 9,11)(10,15)$
$ 6, 2, 2, 2, 1, 1, 1 $ $540$ $6$ $( 1,14, 6, 9,11, 4)( 5,10)( 7,12)( 8,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $135$ $2$ $( 1, 4)( 2, 5)( 6, 9)( 7,10)(11,14)(12,15)$
$ 6, 6, 1, 1, 1 $ $270$ $6$ $( 1,14,11, 9, 6, 4)( 2,10, 7,15,12, 5)$
$ 6, 3, 2, 2, 2 $ $540$ $6$ $( 1, 4)( 2, 5,12,15, 7,10)( 3, 8,13)( 6, 9)(11,14)$
$ 6, 6, 3 $ $270$ $6$ $( 1, 9, 6,14,11, 4)( 2,15, 7, 5,12,10)( 3, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $135$ $2$ $( 1, 4)( 2, 5)( 6,14)( 7,15)( 8,13)( 9,11)(10,12)$
$ 6, 6, 2, 1 $ $540$ $6$ $( 1,14, 6, 9,11, 4)( 2,10,12,15, 7, 5)( 8,13)$
$ 6, 2, 2, 2, 2, 1 $ $540$ $6$ $( 1, 4)( 2, 5,12,10, 7,15)( 3, 8)( 6,14)( 9,11)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $180$ $3$ $( 1, 4, 7)( 2,11,14)( 6, 9,12)$
$ 9, 3, 1, 1, 1 $ $720$ $9$ $( 1,14, 2,11, 9,12, 6, 4, 7)( 5,10,15)$
$ 9, 3, 3 $ $360$ $9$ $( 1,14,12, 6, 4, 2,11, 9, 7)( 3, 8,13)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $360$ $3$ $( 1, 9, 7)( 2,11, 4)( 3, 8,13)( 5,15,10)( 6,14,12)$
$ 6, 3, 2, 2, 1, 1 $ $1620$ $6$ $( 1, 4,12,11, 9, 7)( 2, 6,14)( 8,13)(10,15)$
$ 3, 3, 3, 2, 2, 2 $ $540$ $6$ $( 1, 4, 7)( 2,11,14)( 3,15)( 5, 8)( 6, 9,12)(10,13)$
$ 9, 6 $ $1080$ $18$ $( 1,14, 2,11, 9,12, 6, 4, 7)( 3, 5, 8,10,13,15)$
$ 6, 6, 3 $ $1080$ $6$ $( 1, 4,12,11, 9, 7)( 2, 6,14)( 3,10, 8, 5,13,15)$
$ 6, 3, 2, 2, 2 $ $540$ $6$ $( 1,14, 2, 6, 9, 7)( 3,15)( 4,12,11)( 5,13)( 8,10)$
$ 4, 4, 4, 1, 1, 1 $ $810$ $4$ $( 1, 4, 7,10)( 2, 5,11,14)( 6, 9,12,15)$
$ 12, 3 $ $1620$ $12$ $( 1, 4, 2, 5,11,14,12,15, 6, 9, 7,10)( 3, 8,13)$
$ 4, 4, 4, 2, 1 $ $810$ $4$ $( 1, 4,12,10)( 2, 5, 6,14)( 7,15,11, 9)( 8,13)$
$ 12, 2, 1 $ $1620$ $12$ $( 1, 4, 7,15,11, 9, 2, 5, 6,14,12,10)( 3, 8)$
$ 5, 5, 5 $ $1944$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 10, 5 $ $1944$ $10$ $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $19440=2^{4} \cdot 3^{5} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  19440.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);