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Properties

Label 10T39
10T39 1 3 1->3 2 4 2->4 10 2->10 5 3->5 6 4->6 7 5->7 5->7 5->10 8 6->8 9 7->9 8->10 9->1 10->2
Degree $10$
Order $3840$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2 \wr S_5$

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Copy content magma:G := TransitiveGroup(10, 39);
 

Group invariants

Abstract group:  $C_2 \wr S_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $3840=2^{8} \cdot 3 \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $10$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{5}]S(5)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(5,10)$, $(2,10)(5,7)$, $(1,3,5,7,9)(2,4,6,8,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$120$:  $S_5$
$240$:  $S_5\times C_2$
$1920$:  $(C_2^4:A_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $S_5$

Low degree siblings

10T39, 20T275, 20T279 x 2, 20T285 x 2, 20T288 x 2, 20T289 x 2, 30T517 x 2, 30T524 x 2, 32T206825 x 2, 40T2728 x 2, 40T2731 x 2, 40T2748, 40T2749, 40T2757 x 2, 40T2771 x 2, 40T2772 x 2, 40T2773 x 2, 40T2774 x 2, 40T2779 x 2, 40T2780 x 2, 40T2781 x 2, 40T2782 x 2, 40T2798, 40T2839 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{5}$ $1$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B $2^{4},1^{2}$ $5$ $2$ $4$ $(1,6)(2,7)(3,8)(4,9)$
2C $2,1^{8}$ $5$ $2$ $1$ $(1,6)$
2D $2^{2},1^{6}$ $10$ $2$ $2$ $(3,8)(4,9)$
2E $2^{3},1^{4}$ $10$ $2$ $3$ $( 1, 6)( 4, 9)( 5,10)$
2F $2^{5}$ $20$ $2$ $5$ $( 1, 3)( 2, 7)( 4, 9)( 5,10)( 6, 8)$
2G $2^{2},1^{6}$ $20$ $2$ $2$ $( 2,10)( 5, 7)$
2H $2^{3},1^{4}$ $60$ $2$ $3$ $( 3, 4)( 5,10)( 8, 9)$
2I $2^{4},1^{2}$ $60$ $2$ $4$ $( 2, 3)( 4, 9)( 5,10)( 7, 8)$
2J $2^{5}$ $60$ $2$ $5$ $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$
2K $2^{4},1^{2}$ $60$ $2$ $4$ $( 1,10)( 3, 4)( 5, 6)( 8, 9)$
3A $3^{2},1^{4}$ $80$ $3$ $4$ $(2,9,3)(4,8,7)$
4A $4,1^{6}$ $20$ $4$ $3$ $( 1,10, 6, 5)$
4B $4,2^{3}$ $20$ $4$ $6$ $( 1, 7, 6, 2)( 3, 8)( 4, 9)( 5,10)$
4C $4,2,1^{4}$ $60$ $4$ $4$ $( 3, 4, 8, 9)( 5,10)$
4D $4,2^{2},1^{2}$ $60$ $4$ $5$ $( 2, 3, 7, 8)( 4, 9)( 5,10)$
4E $4^{2},2$ $60$ $4$ $7$ $( 1, 2, 6, 7)( 3, 4, 8, 9)( 5,10)$
4F $4^{2},1^{2}$ $60$ $4$ $6$ $( 1, 8, 6, 3)( 2, 5, 7,10)$
4G $4,2^{3}$ $120$ $4$ $6$ $( 1, 6)( 2, 5, 7,10)( 3, 9)( 4, 8)$
4H $4,2^{2},1^{2}$ $120$ $4$ $5$ $( 1, 2, 6, 7)( 3, 5)( 8,10)$
4I $4^{2},2$ $240$ $4$ $7$ $( 1, 8,10, 9)( 2, 7)( 3, 5, 4, 6)$
4J $4^{2},1^{2}$ $240$ $4$ $6$ $( 1, 9, 5, 2)( 4,10, 7, 6)$
5A $5^{2}$ $384$ $5$ $8$ $( 1,10, 9, 8, 2)( 3, 7, 6, 5, 4)$
6A $3^{2},2^{2}$ $80$ $6$ $6$ $( 1, 6)( 2, 3, 9)( 4, 7, 8)( 5,10)$
6B $6,2^{2}$ $80$ $6$ $7$ $( 1, 7, 9, 6, 2, 4)( 3, 8)( 5,10)$
6C $6,1^{4}$ $80$ $6$ $5$ $( 1, 5, 9, 6,10, 4)$
6D $6,2,1^{2}$ $160$ $6$ $6$ $( 1, 6)( 3,10, 4, 8, 5, 9)$
6E $3^{2},2,1^{2}$ $160$ $6$ $5$ $(1,6)(2,8,9)(3,4,7)$
6F $6,2^{2}$ $160$ $6$ $7$ $( 1, 3)( 2, 4,10, 7, 9, 5)( 6, 8)$
6G $3^{2},2^{2}$ $160$ $6$ $6$ $( 1, 9, 8)( 2,10)( 3, 6, 4)( 5, 7)$
8A $8,1^{2}$ $240$ $8$ $7$ $( 1, 5, 8, 7, 6,10, 3, 2)$
8B $8,2$ $240$ $8$ $8$ $( 1, 4, 2,10, 6, 9, 7, 5)( 3, 8)$
10A $10$ $384$ $10$ $9$ $( 1, 3,10, 7, 9, 6, 8, 5, 2, 4)$
12A $4,3^{2}$ $160$ $12$ $7$ $( 1, 5, 6,10)( 2, 9, 3)( 4, 8, 7)$
12B $6,4$ $160$ $12$ $8$ $( 1, 2, 6, 7)( 3, 4, 5, 8, 9,10)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} + x^{2} + t$ Copy content Toggle raw display