# Properties

 Label 10T39 Degree $10$ Order $3840$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $C_2 \wr S_5$

# Related objects

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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 2, 7)( 5,10)$ $2, 2, 2, 1, 1, 1, 1$ $10$ $2$ $( 2, 7)( 4, 9)( 5,10)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$ $2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $20$ $2$ $( 4,10)( 5, 9)$ $4, 1, 1, 1, 1, 1, 1$ $20$ $4$ $( 4, 5, 9,10)$ $2, 2, 2, 1, 1, 1, 1$ $60$ $2$ $( 2, 7)( 4,10)( 5, 9)$ $4, 2, 1, 1, 1, 1$ $60$ $4$ $( 2, 7)( 4, 5, 9,10)$ $2, 2, 2, 2, 1, 1$ $60$ $2$ $( 1, 6)( 2, 7)( 4,10)( 5, 9)$ $4, 2, 2, 1, 1$ $60$ $4$ $( 1, 6)( 2, 7)( 4, 5, 9,10)$ $2, 2, 2, 2, 2$ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4,10)( 5, 9)$ $4, 2, 2, 2$ $20$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 4, 5, 9,10)$ $3, 3, 1, 1, 1, 1$ $80$ $3$ $( 3, 9, 5)( 4,10, 8)$ $6, 1, 1, 1, 1$ $80$ $6$ $( 3, 9,10, 8, 4, 5)$ $3, 3, 2, 1, 1$ $160$ $6$ $( 2, 7)( 3, 9, 5)( 4,10, 8)$ $6, 2, 1, 1$ $160$ $6$ $( 2, 7)( 3, 9,10, 8, 4, 5)$ $3, 3, 2, 2$ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9, 5)( 4,10, 8)$ $6, 2, 2$ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9,10, 8, 4, 5)$ $2, 2, 2, 2, 1, 1$ $60$ $2$ $( 2, 8)( 3, 7)( 4,10)( 5, 9)$ $4, 2, 2, 1, 1$ $120$ $4$ $( 2, 8)( 3, 7)( 4, 5, 9,10)$ $4, 4, 1, 1$ $60$ $4$ $( 2, 8, 7, 3)( 4, 5, 9,10)$ $2, 2, 2, 2, 2$ $60$ $2$ $( 1, 6)( 2, 8)( 3, 7)( 4,10)( 5, 9)$ $4, 2, 2, 2$ $120$ $4$ $( 1, 6)( 2, 8)( 3, 7)( 4, 5, 9,10)$ $4, 4, 2$ $60$ $4$ $( 1, 6)( 2, 8, 7, 3)( 4, 5, 9,10)$ $4, 4, 1, 1$ $240$ $4$ $( 2, 8, 4,10)( 3, 9, 5, 7)$ $8, 1, 1$ $240$ $8$ $( 2, 8, 4, 5, 7, 3, 9,10)$ $4, 4, 2$ $240$ $4$ $( 1, 6)( 2, 8, 4,10)( 3, 9, 5, 7)$ $8, 2$ $240$ $8$ $( 1, 6)( 2, 8, 4, 5, 7, 3, 9,10)$ $3, 3, 2, 2$ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9, 5)( 4,10, 8)$ $6, 2, 2$ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9,10, 8, 4, 5)$ $4, 3, 3$ $160$ $12$ $( 1, 2, 6, 7)( 3, 9, 5)( 4,10, 8)$ $6, 4$ $160$ $12$ $( 1, 2, 6, 7)( 3, 9,10, 8, 4, 5)$ $5, 5$ $384$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$ $10$ $384$ $10$ $( 1, 7, 3, 9,10, 6, 2, 8, 4, 5)$

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);