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Properties

Label 15T7
15T7 1 2 1->2 4 1->4 11 1->11 3 2->3 7 2->7 8 2->8 3->4 12 3->12 5 4->5 14 4->14 6 5->6 10 5->10 6->7 9 6->9 7->8 13 7->13 8->9 8->13 9->10 10->11 11->12 11->14 12->13 13->14 15 14->15 15->1
Degree $15$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times S_3$

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Copy content magma:G := TransitiveGroup(15, 7);
 

Group invariants

Abstract group:  $D_5\times S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $60=2^{2} \cdot 3 \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $15$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $D(5)[x]S(3)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,4)(2,8)(3,12)(6,9)(7,13)(11,14)$, $(1,11)(2,7)(4,14)(5,10)(8,13)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$10$:  $D_{5}$
$12$:  $D_{6}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Low degree siblings

30T8, 30T10, 30T13

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

Malle's constant $a(G)$:     $1/5$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3 15A1 15A2
Size 1 3 5 15 2 2 2 10 6 6 4 4
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2 15A2 15A1
3 P 1A 2A 2B 2C 1A 5A2 5A1 2B 10A3 10A1 5A2 5A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A 3A 3A
Type
60.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1
60.8.1b R 1 −1 −1 1 1 1 1 −1 −1 −1 1 1
60.8.1c R 1 −1 1 −1 1 1 1 1 −1 −1 1 1
60.8.1d R 1 1 −1 −1 1 1 1 −1 1 1 1 1
60.8.2a R 2 0 2 0 −1 2 2 −1 0 0 −1 −1
60.8.2b R 2 0 −2 0 −1 2 2 1 0 0 −1 −1
60.8.2c1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
60.8.2c2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
60.8.2d1 R 2 −2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
60.8.2d2 R 2 −2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
60.8.4a1 R 4 0 0 0 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
60.8.4a2 R 4 0 0 0 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed