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Properties

Label 15T4
Degree $15$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_5$

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Show commands: Magma

magma: G := TransitiveGroup(15, 4);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_5$
CHM label:   $5[x]S(3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $C_5$

Low degree siblings

30T2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{15}$ $1$ $1$ $0$ $()$
2A $2^{5},1^{5}$ $3$ $2$ $5$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
3A $3^{5}$ $2$ $3$ $10$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
5A1 $5^{3}$ $1$ $5$ $12$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
5A-1 $5^{3}$ $1$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
5A2 $5^{3}$ $1$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
5A-2 $5^{3}$ $1$ $5$ $12$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
10A1 $10,5$ $3$ $10$ $13$ $( 1,13,10, 7, 4)( 2, 9,11, 3, 5,12,14, 6, 8,15)$
10A-1 $10,5$ $3$ $10$ $13$ $( 1, 7,13, 4,10)( 2, 3,14,15,11,12, 8, 9, 5, 6)$
10A3 $10,5$ $3$ $10$ $13$ $( 1, 4, 7,10,13)( 2,15, 8, 6,14,12, 5, 3,11, 9)$
10A-3 $10,5$ $3$ $10$ $13$ $( 1,10, 4,13, 7)( 2, 6, 5, 9, 8,12,11,15,14, 3)$
15A1 $15$ $2$ $15$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
15A-1 $15$ $2$ $15$ $14$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$
15A2 $15$ $2$ $15$ $14$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$
15A-2 $15$ $2$ $15$ $14$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$

Malle's constant $a(G)$:     $1/5$

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:  30.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2
Size 1 3 2 1 1 1 1 3 3 3 3 2 2 2 2
2 P 1A 1A 3A 5A2 5A-2 5A-1 5A1 5A-1 5A2 5A1 5A-2 15A1 15A-1 15A-2 15A2
3 P 1A 2A 1A 5A-2 5A2 5A1 5A-1 10A1 10A3 10A-1 10A-3 5A-2 5A2 5A-1 5A1
5 P 1A 2A 3A 1A 1A 1A 1A 2A 2A 2A 2A 3A 3A 3A 3A
Type
30.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
30.1.1b R 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
30.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
30.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52
30.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
30.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5
30.1.1d1 C 1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
30.1.1d2 C 1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−1 ζ5 ζ5−2 ζ52
30.1.1d3 C 1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
30.1.1d4 C 1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ52 ζ5−2 ζ5−1 ζ5
30.1.2a R 2 0 −1 2 2 2 2 0 0 0 0 −1 −1 −1 −1
30.1.2b1 C 2 0 −1 2ζ5−2 2ζ52 2ζ5 2ζ5−1 0 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2
30.1.2b2 C 2 0 −1 2ζ52 2ζ5−2 2ζ5−1 2ζ5 0 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52
30.1.2b3 C 2 0 −1 2ζ5−1 2ζ5 2ζ5−2 2ζ52 0 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1
30.1.2b4 C 2 0 −1 2ζ5 2ζ5−1 2ζ52 2ζ5−2 0 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5

magma: CharacterTable(G);