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Properties

Label 20T279
Degree $20$
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(20, 279);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $279$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr S_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,20)(4,19)(9,14)(10,13), (1,5,9,14,18)(2,6,10,13,17)(3,7,12,16,20)(4,8,11,15,19), (1,2)(3,4)(5,6)(7,8)(9,19)(10,20)(11,12)(13,14)(15,16)(17,18)
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: $C_2$

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5\times C_2$, $(C_2^4:A_5) : C_2$, $C_2 \wr S_5$

Low degree siblings

10T39 x 2, 20T275, 20T279, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7,18)( 8,17)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1,12)( 2,11)( 3,14)( 4,13)( 7,18)( 8,17)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,17)( 8,18)( 9,19)(10,20)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 1, 5)( 2, 6)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1, 5)( 2, 6)( 7,18)( 8,17)( 9,20)(10,19)(11,15)(12,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $60$ $4$ $( 1, 5,12,16)( 2, 6,11,15)( 9,20)(10,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 5,12,16)( 2, 6,11,15)( 3,14)( 4,13)( 7,18)( 8,17)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,19)(10,20)(11,16)(12,15)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 5)( 3,13)( 4,14)( 7,17)( 8,18)( 9,19)(10,20)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 6,12,15)( 2, 5,11,16)( 3, 4)( 7, 8)( 9,10)(13,14)(17,18)(19,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $60$ $4$ $( 1, 6,12,15)( 2, 5,11,16)( 3, 4)( 7,17)( 8,18)( 9,19)(10,20)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1, 5)( 2, 6)( 3,20)( 4,19)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1, 5)( 2, 6)( 3, 9,14,20)( 4,10,13,19)( 7,18)( 8,17)(11,15)(12,16)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $60$ $4$ $( 1, 5,12,16)( 2, 6,11,15)( 3, 9,14,20)( 4,10,13,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1, 6)( 2, 5)( 3,10,14,19)( 4, 9,13,20)( 7, 8)(11,16)(12,15)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 5)( 3,19)( 4,20)( 7,17)( 8,18)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1, 6,12,15)( 2, 5,11,16)( 3,10,14,19)( 4, 9,13,20)( 7,17)( 8,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1, 5, 9)( 2, 6,10)(11,15,19)(12,16,20)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $160$ $6$ $( 1, 5,20,12,16, 9)( 2, 6,19,11,15,10)( 7,18)( 8,17)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $80$ $6$ $( 1, 5, 9)( 2, 6,10)( 3,14)( 4,13)( 7,18)( 8,17)(11,15,19)(12,16,20)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1, 6,20,11,16,10)( 2, 5,19,12,15, 9)( 3, 4)( 7, 8)(13,14)(17,18)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 4)( 7,17)( 8,18)(11,16,19,12,15,20)(13,14)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1, 6,20,11,16,10)( 2, 5,19,12,15, 9)( 3,13)( 4,14)( 7,17)( 8,18)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $160$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 7)( 4, 8)(11,15,19)(12,16,20)(13,17)(14,18)$
$ 6, 6, 4, 4 $ $160$ $12$ $( 1, 5,20,12,16, 9)( 2, 6,19,11,15,10)( 3,18,14, 7)( 4,17,13, 8)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1, 6,20,11,16,10)( 2, 5,19,12,15, 9)( 3, 8)( 4, 7)(13,18)(14,17)$
$ 6, 6, 4, 4 $ $160$ $12$ $( 1, 6, 9, 2, 5,10)( 3,17,14, 8)( 4,18,13, 7)(11,16,19,12,15,20)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $240$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3,12,16,20)( 4,11,15,19)$
$ 8, 8, 2, 2 $ $240$ $8$ $( 1, 5,20, 3,12,16, 9,14)( 2, 6,19, 4,11,15,10,13)( 7,18)( 8,17)$
$ 8, 8, 2, 2 $ $240$ $8$ $( 1, 6,20, 4,12,15, 9,13)( 2, 5,19, 3,11,16,10,14)( 7, 8)(17,18)$
$ 4, 4, 4, 4, 2, 2 $ $240$ $4$ $( 1, 6, 9,13)( 2, 5,10,14)( 3,11,16,19)( 4,12,15,20)( 7,17)( 8,18)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 7,12,16,20)( 4, 8,11,15,19)$
$ 10, 10 $ $384$ $10$ $( 1, 6,20, 4, 7,11,16,10,14,17)( 2, 5,19, 3, 8,12,15, 9,13,18)$

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);
 
Nilpotency class:   not nilpotent
Label:  3840.ch
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);