Properties

Label 15T7
Degree $15$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 7);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times S_3$
CHM label:  $D(5)[x]S(3)$
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)(3,12)(6,9)(7,13)(11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$10$:  $D_{5}$
$12$:  $D_{6}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Low degree siblings

30T8, 30T10, 30T13

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.8
magma: IdentifyGroup(G);
 
Character table:   
      2  2  2  2  2   .  1   1   .   1  1  1  1
      3  1  1  .  .   1  1   .   1   .  1  1  1
      5  1  .  1  .   1  .   1   1   1  1  1  1

        1a 2a 2b 2c 15a 6a 10a 15b 10b 5a 3a 5b
     2P 1a 1a 1a 1a 15b 3a  5a 15a  5b 5b 3a 5a
     3P 1a 2a 2b 2c  5a 2a 10b  5b 10a 5b 1a 5a
     5P 1a 2a 2b 2c  3a 6a  2b  3a  2b 1a 3a 1a
     7P 1a 2a 2b 2c 15b 6a 10b 15a 10a 5b 3a 5a
    11P 1a 2a 2b 2c 15a 6a 10a 15b 10b 5a 3a 5b
    13P 1a 2a 2b 2c 15b 6a 10b 15a 10a 5b 3a 5a

X.1      1  1  1  1   1  1   1   1   1  1  1  1
X.2      1 -1 -1  1   1 -1  -1   1  -1  1  1  1
X.3      1 -1  1 -1   1 -1   1   1   1  1  1  1
X.4      1  1 -1 -1   1  1  -1   1  -1  1  1  1
X.5      2 -2  .  .  -1  1   .  -1   .  2 -1  2
X.6      2  2  .  .  -1 -1   .  -1   .  2 -1  2
X.7      2  . -2  .   A  .  -A  *A -*A *A  2  A
X.8      2  . -2  .  *A  . -*A   A  -A  A  2 *A
X.9      2  .  2  .   A  .   A  *A  *A *A  2  A
X.10     2  .  2  .  *A  .  *A   A   A  A  2 *A
X.11     4  .  .  .  -A  .   . -*A   .  B -2 *B
X.12     4  .  .  . -*A  .   .  -A   . *B -2  B

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)+2*E(5)^4
  = -1+Sqrt(5) = 2b5

magma: CharacterTable(G);