↑

Properties

Label 15T8
Degree $15$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5\times C_3$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(15, 8);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_5\times C_3$
CHM label:   $F(5)[x]3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,4,13)(2,14,8,11)(3,6,12,9), (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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $F_5$

Low degree siblings

30T7

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  60.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 5A 6A1 6A-1 12A1 12A-1 12A5 12A-5 15A1 15A-1
Size 1 5 1 1 5 5 4 5 5 5 5 5 5 4 4
2 P 1A 1A 3A-1 3A1 2A 2A 5A 3A1 3A-1 6A-1 6A1 6A1 6A-1 15A-1 15A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 2A 2A 4A-1 4A1 4A-1 4A1 5A 5A
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 6A-1 6A1 12A-5 12A5 12A-1 12A1 3A-1 3A1
Type
60.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.6.1b R 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1
60.6.1c1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
60.6.1c2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
60.6.1d1 C 1 −1 1 1 −i i 1 −1 −1 i −i i −i 1 1
60.6.1d2 C 1 −1 1 1 i −i 1 −1 −1 −i i −i i 1 1
60.6.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
60.6.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
60.6.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 −ζ122 ζ124
60.6.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 ζ124 −ζ122
60.6.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 −ζ122 ζ124
60.6.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 ζ124 −ζ122
60.6.4a R 4 0 4 4 0 0 −1 0 0 0 0 0 0 −1 −1
60.6.4b1 C 4 0 4ζ3−1 4ζ3 0 0 −1 0 0 0 0 0 0 −ζ3−1 −ζ3
60.6.4b2 C 4 0 4ζ3 4ζ3−1 0 0 −1 0 0 0 0 0 0 −ζ3 −ζ3−1

magma: CharacterTable(G);