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Properties

Label 10T39
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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Show commands: Magma

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

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \wr S_5$
CHM label:  $[2^{5}]S(5)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (5,10), (2,10)(5,7), (1,3,5,7,9)(2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

|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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 7)( 5,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 4,10)( 5, 9)$
$ 4, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 4, 5, 9,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 2, 7)( 4,10)( 5, 9)$
$ 4, 2, 1, 1, 1, 1 $ $60$ $4$ $( 2, 7)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 1, 6)( 2, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 1, 1 $ $60$ $4$ $( 1, 6)( 2, 7)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4,10)( 5, 9)$
$ 4, 2, 2, 2 $ $20$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 4, 5, 9,10)$
$ 3, 3, 1, 1, 1, 1 $ $80$ $3$ $( 3, 9, 5)( 4,10, 8)$
$ 6, 1, 1, 1, 1 $ $80$ $6$ $( 3, 9,10, 8, 4, 5)$
$ 3, 3, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 9,10, 8, 4, 5)$
$ 3, 3, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9,10, 8, 4, 5)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 2, 8)( 3, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 1, 1 $ $120$ $4$ $( 2, 8)( 3, 7)( 4, 5, 9,10)$
$ 4, 4, 1, 1 $ $60$ $4$ $( 2, 8, 7, 3)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 8)( 3, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 2 $ $120$ $4$ $( 1, 6)( 2, 8)( 3, 7)( 4, 5, 9,10)$
$ 4, 4, 2 $ $60$ $4$ $( 1, 6)( 2, 8, 7, 3)( 4, 5, 9,10)$
$ 4, 4, 1, 1 $ $240$ $4$ $( 2, 8, 4,10)( 3, 9, 5, 7)$
$ 8, 1, 1 $ $240$ $8$ $( 2, 8, 4, 5, 7, 3, 9,10)$
$ 4, 4, 2 $ $240$ $4$ $( 1, 6)( 2, 8, 4,10)( 3, 9, 5, 7)$
$ 8, 2 $ $240$ $8$ $( 1, 6)( 2, 8, 4, 5, 7, 3, 9,10)$
$ 3, 3, 2, 2 $ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 2 $ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9,10, 8, 4, 5)$
$ 4, 3, 3 $ $160$ $12$ $( 1, 2, 6, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 4 $ $160$ $12$ $( 1, 2, 6, 7)( 3, 9,10, 8, 4, 5)$
$ 5, 5 $ $384$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
$ 10 $ $384$ $10$ $( 1, 7, 3, 9,10, 6, 2, 8, 4, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3840=2^{8} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  3840.ch
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);