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Properties

Label 20T288
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, 288);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $288$
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:  (1,7)(2,8)(3,16,10,4,15,9)(5,19,13,6,20,14)(11,17)(12,18), (1,14,5,2,13,6)(3,16,11,4,15,12)(7,17)(8,18)(9,10)(19,20)
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 x 2, 20T285 x 2, 20T288, 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 $ $1$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7, 8)( 9,10)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,10)(11,12)(13,14)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 7)( 2, 8)( 3, 4)( 5, 6)( 9,10)(11,17)(12,18)(13,14)(15,16)(19,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $60$ $4$ $( 1,18,12, 7)( 2,17,11, 8)( 3, 4)( 5, 6)( 9,19)(10,20)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,18)( 2,17)( 3,13)( 4,14)( 5, 6)( 7,12)( 8,11)( 9,19)(10,20)(15,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 7,12,18)( 2, 8,11,17)( 3,13)( 4,14)( 5,15)( 6,16)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1,17)( 2,18)( 3,14)( 4,13)( 5,16)( 6,15)( 7,11)( 8,12)( 9,20)(10,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $4$ $( 1, 8,12,17)( 2, 7,11,18)( 3,14)( 4,13)( 5,16)( 6,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $60$ $2$ $( 1, 8)( 2, 7)( 5,16)( 6,15)(11,18)(12,17)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 1,17,12, 8)( 2,18,11, 7)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1, 8)( 2, 7)( 3,15)( 4,16)( 5,13)( 6,14)(11,18)(12,17)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,17,12, 8)( 2,18,11, 7)( 3,15)( 4,16)( 5,13)( 6,14)( 9,20)(10,19)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $60$ $4$ $( 1, 8,12,17)( 2, 7,11,18)( 3,15,14, 6)( 4,16,13, 5)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,18)( 2,17)( 3, 5)( 4, 6)( 7,12)( 8,11)( 9,19)(10,20)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1, 7,12,18)( 2, 8,11,17)( 3, 5)( 4, 6)( 9,10)(13,15)(14,16)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1,18,12, 7)( 2,17,11, 8)( 3, 5,14,16)( 4, 6,13,15)( 9,19)(10,20)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1, 8, 4)( 2, 7, 3)(11,18,14)(12,17,13)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $160$ $6$ $( 1,17,13,12, 8, 4)( 2,18,14,11, 7, 3)( 9,20)(10,19)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $80$ $6$ $( 1, 8, 4)( 2, 7, 3)( 5,16)( 6,15)( 9,20)(10,19)(11,18,14)(12,17,13)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,18, 4,11, 8,14)( 2,17, 3,12, 7,13)( 5,15)( 6,16)( 9,19)(10,20)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1, 7,13, 2, 8,14)( 3,12,18, 4,11,17)( 5,15)( 6,16)( 9,10)(19,20)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,18, 4,11, 8,14)( 2,17, 3,12, 7,13)( 5, 6)( 9,10)(15,16)(19,20)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1, 8,13,12,17, 4)( 2, 7,14,11,18, 3)( 5,20)( 6,19)( 9,16)(10,15)$
$ 4, 4, 3, 3, 3, 3 $ $160$ $12$ $( 1,17, 4)( 2,18, 3)( 5, 9,16,20)( 6,10,15,19)( 7,14,11)( 8,13,12)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1,18,13, 2,17,14)( 3,12, 7, 4,11, 8)( 5,10)( 6, 9)(15,20)(16,19)$
$ 6, 6, 4, 4 $ $160$ $12$ $( 1, 7, 4,11,17,14)( 2, 8, 3,12,18,13)( 5,19,16,10)( 6,20,15, 9)$
$ 8, 8, 1, 1, 1, 1 $ $240$ $8$ $( 1, 8,13, 5,12,17, 4,16)( 2, 7,14, 6,11,18, 3,15)$
$ 4, 4, 4, 4, 2, 2 $ $240$ $4$ $( 1,17, 4,16)( 2,18, 3,15)( 5,12, 8,13)( 6,11, 7,14)( 9,20)(10,19)$
$ 8, 8, 2, 2 $ $240$ $8$ $( 1,18,13,15,12, 7, 4, 6)( 2,17,14,16,11, 8, 3, 5)( 9,19)(10,20)$
$ 4, 4, 4, 4, 2, 2 $ $240$ $4$ $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,10)(11,17,14,16)(12,18,13,15)(19,20)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1, 8, 4,16, 9)( 2, 7, 3,15,10)( 5,20,12,17,13)( 6,19,11,18,14)$
$ 10, 10 $ $384$ $10$ $( 1,18, 4, 6, 9,11, 8,14,16,19)( 2,17, 3, 5,10,12, 7,13,15,20)$

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);