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Properties

Label 15T50
15T50 1 6 1->6 2 7 2->7 3 3->6 8 3->8 12 3->12 4 9 4->9 5 10 5->10 6->9 6->9 11 6->11 7->12 13 8->13 9->12 14 9->14 15 10->15 11->1 12->2 12->15 13->3 14->4 15->3 15->5
Degree $15$
Order $3000$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\wr C_3$

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Copy content magma:G := TransitiveGroup(15, 50);
 

Group invariants

Abstract group:  $D_5\wr C_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $3000=2^{3} \cdot 3 \cdot 5^{3}$
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 $t$:  $50$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[D(5)^{3}]3=D(5)wr3$
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:  $(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)$, $(3,12)(6,9)$, $(3,6,9,12,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

20T269, 30T408, 30T418, 30T419 x 2, 30T420 x 2, 30T421, 30T425, 30T426, 40T2384

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 $2^{2},1^{11}$ $15$ $2$ $2$ $( 2, 5)( 8,14)$
2B $2^{4},1^{7}$ $75$ $2$ $4$ $( 1,13)( 2,14)( 4,10)( 5,11)$
2C $2^{6},1^{3}$ $125$ $2$ $6$ $( 1, 4)( 2,14)( 3,12)( 5,11)( 6, 9)( 7,13)$
3A1 $3^{5}$ $100$ $3$ $10$ $( 1, 9, 8)( 2,10, 3)( 4,12,11)( 5,13, 6)( 7,15,14)$
3A-1 $3^{5}$ $100$ $3$ $10$ $( 1, 8, 9)( 2, 3,10)( 4,11,12)( 5, 6,13)( 7,14,15)$
5A1 $5,1^{10}$ $6$ $5$ $4$ $( 1, 4, 7,10,13)$
5A2 $5,1^{10}$ $6$ $5$ $4$ $( 1, 7,13, 4,10)$
5B1 $5^{3}$ $8$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
5B2 $5^{3}$ $8$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
5C1 $5^{2},1^{5}$ $12$ $5$ $8$ $( 1, 7,13, 4,10)( 3, 6, 9,12,15)$
5C2 $5^{2},1^{5}$ $12$ $5$ $8$ $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$
5D1 $5^{2},1^{5}$ $12$ $5$ $8$ $( 1,10, 4,13, 7)( 2, 8,14, 5,11)$
5D2 $5^{2},1^{5}$ $12$ $5$ $8$ $( 1, 4, 7,10,13)( 2,14,11, 8, 5)$
5E1 $5^{3}$ $24$ $5$ $12$ $( 1, 4, 7,10,13)( 2,14,11, 8, 5)( 3, 9,15, 6,12)$
5E2 $5^{3}$ $24$ $5$ $12$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
6A1 $6^{2},3$ $500$ $6$ $12$ $( 1,14, 9, 4, 2, 6)( 3,13,11,12, 7, 5)( 8,15,10)$
6A-1 $6^{2},3$ $500$ $6$ $12$ $( 1, 6, 2, 4, 9,14)( 3, 5, 7,12,11,13)( 8,10,15)$
10A1 $5,2^{2},1^{6}$ $30$ $10$ $6$ $( 1,10, 4,13, 7)( 2, 5)( 8,14)$
10A3 $5,2^{2},1^{6}$ $30$ $10$ $6$ $( 1,13,10, 7, 4)( 2, 5)( 8,14)$
10B1 $5,2^{2},1^{6}$ $30$ $10$ $6$ $( 1,13,10, 7, 4)( 3,15)( 6,12)$
10B3 $5,2^{2},1^{6}$ $30$ $10$ $6$ $( 2, 5)( 3, 9,15, 6,12)( 8,14)$
10C1 $5^{2},2^{2},1$ $60$ $10$ $10$ $( 1, 4, 7,10,13)( 2,14)( 3,12, 6,15, 9)( 5,11)$
10C3 $5^{2},2^{2},1$ $60$ $10$ $10$ $( 1,10, 4,13, 7)( 2,14)( 3,15,12, 9, 6)( 5,11)$
10D1 $5^{2},2^{2},1$ $60$ $10$ $10$ $( 1,13,10, 7, 4)( 2, 5, 8,11,14)( 3, 6)( 9,15)$
10D3 $5^{2},2^{2},1$ $60$ $10$ $10$ $( 1, 7,13, 4,10)( 2,11, 5,14, 8)( 3, 6)( 9,15)$
10E1 $5,2^{4},1^{2}$ $150$ $10$ $8$ $( 1,13)( 2,14)( 3, 9,15, 6,12)( 4,10)( 5,11)$
10E3 $5,2^{4},1^{2}$ $150$ $10$ $8$ $( 1,13)( 2,14)( 3, 6, 9,12,15)( 4,10)( 5,11)$
15A1 $15$ $200$ $15$ $14$ $( 1, 2,12,13,14, 9,10,11, 6, 7, 8, 3, 4, 5,15)$
15A-1 $15$ $200$ $15$ $14$ $( 1,15, 5, 4, 3, 8, 7, 6,11,10, 9,14,13,12, 2)$
15A2 $15$ $200$ $15$ $14$ $( 1,12,14,10, 6, 8, 4,15, 2,13, 9,11, 7, 3, 5)$
15A-2 $15$ $200$ $15$ $14$ $( 1, 5, 3, 7,11, 9,13, 2,15, 4, 8, 6,10,14,12)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed