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Properties

Label 15T3
Degree $15$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times C_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 3);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times C_3$
CHM label:   $D(5)[x]3$
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:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $D_{5}$

Low degree siblings

30T4

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

Malle's constant $a(G)$:     $1/6$

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:  30.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 5A1 5A2 6A1 6A-1 15A1 15A-1 15A2 15A-2
Size 1 5 1 1 2 2 5 5 2 2 2 2
2 P 1A 1A 3A-1 3A1 5A2 5A1 3A1 3A-1 15A2 15A-2 15A1 15A-1
3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 5A1 5A1 5A2 5A2
5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 3A1 3A-1 3A-1 3A1
Type
30.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1
30.2.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1
30.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
30.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
30.2.1d1 C 1 −1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3−1 ζ3
30.2.1d2 C 1 −1 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
30.2.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
30.2.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
30.2.2b1 C 2 0 2ζ15−5 2ζ155 ζ15−6+ζ156 ζ15−3+ζ153 0 0 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157
30.2.2b2 C 2 0 2ζ155 2ζ15−5 ζ15−6+ζ156 ζ15−3+ζ153 0 0 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154
30.2.2b3 C 2 0 2ζ15−5 2ζ155 ζ15−3+ζ153 ζ15−6+ζ156 0 0 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157
30.2.2b4 C 2 0 2ζ155 2ζ15−5 ζ15−3+ζ153 ζ15−6+ζ156 0 0 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155

magma: CharacterTable(G);