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Properties

Label 15T2
Degree $15$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(15, 2);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{15}$
CHM label:  $D(15)$
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:  (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8), (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)
$2$:  $C_2$
$6$:  $S_3$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Low degree siblings

30T3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $30=2 \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  30.3
magma: IdentifyGroup(G);
 
Character table:    
     2  1  1   .   .  .   .  .  .   .
     3  1  .   1   1  1   1  1  1   1
     5  1  .   1   1  1   1  1  1   1

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

X.1     1  1   1   1  1   1  1  1   1
X.2     1 -1   1   1  1   1  1  1   1
X.3     2  .  -1  -1  2  -1 -1  2  -1
X.4     2  .   A   D *E   C -1  E   B
X.5     2  .   B   A  E   D -1 *E   C
X.6     2  .   C   B *E   A -1  E   D
X.7     2  .   D   C  E   B -1 *E   A
X.8     2  .   E  *E *E   E  2  E  *E
X.9     2  .  *E   E  E  *E  2 *E   E

A = E(15)^7+E(15)^8
B = E(15)^4+E(15)^11
C = E(15)^2+E(15)^13
D = E(15)+E(15)^14
E = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5

magma: CharacterTable(G);