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Properties

Label 15T33
15T33 1 4 1->4 6 1->6 11 1->11 2 5 2->5 7 2->7 3 3->6 4->7 14 4->14 4->14 8 5->8 10 5->10 9 6->9 6->11 7->10 8->11 13 8->13 9->4 12 9->12 10->13 11->1 11->14 15 12->15 13->1 14->2 14->9 15->3
Degree $15$
Order $810$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:C_{10}$

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Copy content magma:G := TransitiveGroup(15, 33);
 

Group invariants

Abstract group:  $C_3^4:C_{10}$
Copy content magma:IdentifyGroup(G);
 
Order:  $810=2 \cdot 3^{4} \cdot 5$
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 $t$:  $33$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[3^{4}:2]5$
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:  $(1,6,11)(4,14,9)$, $(1,11)(2,7)(4,14)(5,10)(8,13)$, $(1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $C_5$

Low degree siblings

15T33 x 7, 30T190 x 8, 45T118 x 2, 45T119 x 16, 45T120 x 8

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 $2^{5},1^{5}$ $81$ $2$ $5$ $( 1, 6)( 2,12)( 3, 8)( 4,14)(10,15)$
3A $3^{4},1^{3}$ $10$ $3$ $8$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
3B $3^{3},1^{6}$ $10$ $3$ $6$ $( 2,12, 7)( 4,14, 9)( 5,15,10)$
3C $3^{3},1^{6}$ $10$ $3$ $6$ $( 2,12, 7)( 3,13, 8)( 4,14, 9)$
3D $3^{5}$ $10$ $3$ $10$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
3E $3^{4},1^{3}$ $10$ $3$ $8$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 5,10,15)$
3F $3^{4},1^{3}$ $10$ $3$ $8$ $( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$
3G $3^{2},1^{9}$ $10$ $3$ $4$ $( 2, 7,12)( 4,14, 9)$
3H $3^{2},1^{9}$ $10$ $3$ $4$ $( 1, 6,11)( 2,12, 7)$
5A1 $5^{3}$ $81$ $5$ $12$ $( 1, 3,10,12,14)( 2, 4, 6, 8,15)( 5, 7, 9,11,13)$
5A-1 $5^{3}$ $81$ $5$ $12$ $( 1,14,12,10, 3)( 2,15, 8, 6, 4)( 5,13,11, 9, 7)$
5A2 $5^{3}$ $81$ $5$ $12$ $( 1,10,14, 3,12)( 2, 6,15, 4, 8)( 5, 9,13, 7,11)$
5A-2 $5^{3}$ $81$ $5$ $12$ $( 1,12, 3,14,10)( 2, 8, 4,15, 6)( 5,11, 7,13, 9)$
10A1 $10,5$ $81$ $10$ $13$ $( 1, 2, 3, 4,10, 6,12, 8,14,15)( 5,11, 7,13, 9)$
10A-1 $10,5$ $81$ $10$ $13$ $( 1,15,14, 8,12, 6,10, 4, 3, 2)( 5, 9,13, 7,11)$
10A3 $10,5$ $81$ $10$ $13$ $( 1, 4,12,15, 3, 6,14, 2,10, 8)( 5,13,11, 9, 7)$
10A-3 $10,5$ $81$ $10$ $13$ $( 1, 8,10, 2,14, 6, 3,15,12, 4)( 5, 7, 9,11,13)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 3B 3C 3D 3E 3F 3G 3H 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3
Size 1 81 10 10 10 10 10 10 10 10 81 81 81 81 81 81 81 81
2 P 1A 1A 3A 3B 3C 3D 3E 3F 3G 3H 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 5A-2 5A2 5A1 5A-1 10A3 10A-3 10A-1 10A1
5 P 1A 2A 3A 3B 3C 3D 3E 3F 3G 3H 1A 1A 1A 1A 2A 2A 2A 2A
Type
810.102.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
810.102.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
810.102.1c1 C 1 1 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−1 ζ5 ζ5−2
810.102.1c2 C 1 1 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ5 ζ5−1 ζ52
810.102.1c3 C 1 1 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ52 ζ5−2 ζ5−1
810.102.1c4 C 1 1 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5−2 ζ52 ζ5
810.102.1d1 C 1 −1 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ52 −ζ5−1 −ζ5 −ζ5−2
810.102.1d2 C 1 −1 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−2 −ζ5 −ζ5−1 −ζ52
810.102.1d3 C 1 −1 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5 −ζ52 −ζ5−2 −ζ5−1
810.102.1d4 C 1 −1 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−1 −ζ5−2 −ζ52 −ζ5
810.102.10a R 10 0 −5 −2 4 −2 1 −2 4 1 0 0 0 0 0 0 0 0
810.102.10b R 10 0 −2 −5 −2 4 1 4 −2 1 0 0 0 0 0 0 0 0
810.102.10c R 10 0 −2 4 −5 −2 −2 1 1 4 0 0 0 0 0 0 0 0
810.102.10d R 10 0 −2 4 1 1 4 −2 −5 −2 0 0 0 0 0 0 0 0
810.102.10e R 10 0 1 1 −2 4 −2 −2 4 −5 0 0 0 0 0 0 0 0
810.102.10f R 10 0 1 1 4 −2 −5 4 −2 −2 0 0 0 0 0 0 0 0
810.102.10g R 10 0 4 −2 −2 −5 4 1 1 −2 0 0 0 0 0 0 0 0
810.102.10h R 10 0 4 −2 1 1 −2 −5 −2 4 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed