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Properties

Label 15T44
Degree $15$
Order $2430$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(15, 44);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^3:C_6$
CHM label:  $[3^{5}:2]5$
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:  (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$
$810$:  15T33

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $C_5$

Low degree siblings

15T44 x 15, 30T394 x 16, 45T253 x 8, 45T254 x 16, 45T255 x 32, 45T256 x 32, 45T257 x 64, 45T258 x 64

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $243$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $162$ $15$ $( 1, 4, 7, 5, 8,11,14, 2,15, 3, 6, 9,12,10,13)$
$ 10, 5 $ $243$ $10$ $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 15 $ $162$ $15$ $( 1, 7,13, 4, 5,11, 2, 8,14,15, 6,12, 3, 9,10)$
$ 10, 5 $ $243$ $10$ $( 1,12, 3,14, 5, 6, 7, 8, 9,10)( 2,13, 4,15,11)$
$ 5, 5, 5 $ $81$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 15 $ $162$ $15$ $( 1,13, 5, 2,14,11, 8,15,12, 9, 6, 3,10, 7, 4)$
$ 10, 5 $ $243$ $10$ $( 1, 8, 5, 2, 9,11,13,15, 7, 4)( 3,10,12,14, 6)$
$ 5, 5, 5 $ $81$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 15 $ $162$ $15$ $( 1, 5,14, 8, 2,11,15, 9, 3,12, 6,10, 4,13, 7)$
$ 10, 5 $ $243$ $10$ $( 1,15,14,13,12,11, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2430=2 \cdot 3^{5} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2430.f
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);