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Properties

Label 15T11
Degree $15$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5 \times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 11);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_5 \times S_3$
CHM label:   $F(5)[x]S(3)$
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:  (1,11)(2,7)(4,14)(5,10)(8,13), (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)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

Low degree siblings

30T23, 30T24, 30T32

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.36
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 10A 12A1 12A-1 15A
Size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 5A 6A 6A 15A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2B 10A 4A1 4A-1 5A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 2A 12A1 12A-1 3A
Type
120.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1b R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1
120.36.1c R 1 −1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 1
120.36.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
120.36.1e1 C 1 −1 −1 1 1 −i i i −i 1 −1 −1 i −i 1
120.36.1e2 C 1 −1 −1 1 1 i −i −i i 1 −1 −1 −i i 1
120.36.1f1 C 1 1 −1 −1 1 −i i −i i 1 −1 1 i −i 1
120.36.1f2 C 1 1 −1 −1 1 i −i i −i 1 −1 1 −i i 1
120.36.2a R 2 0 2 0 −1 2 2 0 0 2 −1 0 −1 −1 −1
120.36.2b R 2 0 2 0 −1 −2 −2 0 0 2 −1 0 1 1 −1
120.36.2c1 C 2 0 −2 0 −1 −2i 2i 0 0 2 1 0 −i i −1
120.36.2c2 C 2 0 −2 0 −1 2i −2i 0 0 2 1 0 i −i −1
120.36.4a R 4 4 0 0 4 0 0 0 0 −1 0 −1 0 0 −1
120.36.4b R 4 −4 0 0 4 0 0 0 0 −1 0 1 0 0 −1
120.36.8a R 8 0 0 0 −4 0 0 0 0 −2 0 0 0 0 1

magma: CharacterTable(G);