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Properties

Label 15T45
Degree $15$
Order $2430$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:D_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(15, 45);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:D_{15}$
CHM label:  $1/2[3^{5}:2]D(5)$
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:  (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$10$:  $D_{5}$
$30$:  $D_{15}$
$810$:  15T34

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T45 x 7, 30T389 x 8, 30T392 x 4, 45T248 x 4, 45T249 x 4, 45T259 x 16, 45T260 x 16, 45T264 x 8

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, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)( 3,13, 8)( 4, 9,14)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3, 8,13)( 4,14, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2, 7,12)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)$
$ 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, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2, 7,12)( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3,13, 8)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2,12, 7)( 3, 8,13)( 4, 9,14)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2,12, 7)( 3,13, 8)$
$ 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 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 5,15,10)$
$ 15 $ $162$ $15$ $( 1, 4, 7,10, 3, 6, 9,12,15, 8,11,14, 2, 5,13)$
$ 15 $ $162$ $15$ $( 1, 7,13, 4,10, 6,12, 3, 9,15,11, 2, 8,14, 5)$
$ 15 $ $162$ $15$ $( 1,13,10,12, 9, 6, 3,15, 2,14,11, 8, 5, 7, 4)$
$ 15 $ $162$ $15$ $( 1,10, 9, 3,12, 6,15,14, 8, 2,11, 5, 4,13, 7)$
$ 6, 6, 2, 1 $ $135$ $6$ $( 2,15,12, 5, 7,10)( 3, 9, 8, 4,13,14)( 6,11)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $135$ $2$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)$
$ 6, 6, 2, 1 $ $135$ $6$ $( 1, 6)( 2, 5, 7,15,12,10)( 3, 9,13,14, 8, 4)$
$ 6, 2, 2, 2, 2, 1 $ $135$ $6$ $( 2, 5,12,10, 7,15)( 3,14)( 4,13)( 6,11)( 8, 9)$
$ 6, 2, 2, 2, 2, 1 $ $135$ $6$ $( 1,11)( 2,15)( 3,14,13, 4, 8, 9)( 5,12)( 7,10)$
$ 6, 6, 2, 1 $ $135$ $6$ $( 1, 6)( 2,10, 7, 5,12,15)( 3,14, 8, 9,13, 4)$
$ 6, 6, 2, 1 $ $135$ $6$ $( 2,10,12,15, 7, 5)( 3, 4,13, 9, 8,14)( 6,11)$
$ 6, 2, 2, 2, 2, 1 $ $135$ $6$ $( 1,11)( 2, 5)( 3, 4, 8,14,13, 9)( 7,15)(10,12)$
$ 6, 2, 2, 2, 2, 1 $ $135$ $6$ $( 1, 6)( 2,15, 7,10,12, 5)( 3, 4)( 8,14)( 9,13)$

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.g
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);