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Properties

Label 20T42
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 42);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4\times F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,18,2,17)(3,16,4,15)(5,13,6,14)(7,11,8,12)(9,19,10,20), (1,2)(3,7,9,5)(4,8,10,6)(11,13,20,18)(12,14,19,17), (1,20,3,14)(2,19,4,13)(5,18,9,16)(6,17,10,15)(7,11,8,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$
$20$:  $F_5$
$32$:  $C_4 \times D_4$
$40$:  $F_{5}\times C_2$ x 3
$80$:  20T16

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T42 x 3, 40T109 x 2, 40T110 x 2, 40T112, 40T118 x 2, 40T119 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 2, 1, 1 $ $10$ $4$ $( 3, 6, 9, 8)( 4, 5,10, 7)(11,17,20,14)(12,18,19,13)(15,16)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $5$ $4$ $( 3, 6, 9, 8)( 4, 5,10, 7)(11,18,20,13)(12,17,19,14)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $5$ $4$ $( 3, 8, 9, 6)( 4, 7,10, 5)(11,13,20,18)(12,14,19,17)$
$ 4, 4, 4, 4, 2, 1, 1 $ $10$ $4$ $( 3, 8, 9, 6)( 4, 7,10, 5)(11,14,20,17)(12,13,19,18)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $10$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,19)(12,20)(13,17)(14,18)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,20)(12,19)(13,18)(14,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $5$ $4$ $( 1, 2)( 3, 5, 9, 7)( 4, 6,10, 8)(11,17,20,14)(12,18,19,13)(15,16)$
$ 4, 4, 4, 4, 2, 2 $ $5$ $4$ $( 1, 2)( 3, 7, 9, 5)( 4, 8,10, 6)(11,14,20,17)(12,13,19,18)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)(11,19)(12,20)(13,17)(14,18)(15,16)$
$ 10, 5, 5 $ $8$ $10$ $( 1, 3, 5, 8,10, 2, 4, 6, 7, 9)(11,13,15,18,20)(12,14,16,17,19)$
$ 10, 10 $ $4$ $10$ $( 1, 3, 5, 8,10, 2, 4, 6, 7, 9)(11,14,15,17,20,12,13,16,18,19)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 4, 5, 7,10)( 2, 3, 6, 8, 9)(11,13,15,18,20)(12,14,16,17,19)$
$ 20 $ $8$ $20$ $( 1,11, 3,14, 5,15, 8,17,10,20, 2,12, 4,13, 6,16, 7,18, 9,19)$
$ 10, 10 $ $8$ $10$ $( 1,11, 4,13, 5,15, 7,18,10,20)( 2,12, 3,14, 6,16, 8,17, 9,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 9,17)( 2,12,10,18)( 3,16, 7,13)( 4,15, 8,14)( 5,20, 6,19)$
$ 4, 4, 4, 4, 2, 2 $ $10$ $4$ $( 1,11,10,18)( 2,12, 9,17)( 3,16, 8,14)( 4,15, 7,13)( 5,20)( 6,19)$
$ 4, 4, 4, 4, 2, 2 $ $10$ $4$ $( 1,11, 5,13)( 2,12, 6,14)( 3,17)( 4,18)( 7,20,10,15)( 8,19, 9,16)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 6,14)( 2,12, 5,13)( 3,17, 4,18)( 7,20, 9,16)( 8,19,10,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,11)( 2,12)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 2,12)( 3,19, 4,20)( 5,18, 6,17)( 7,15, 8,16)( 9,14,10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,11)( 8,12)( 9,14)(10,13)$
$ 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,15, 2,16)( 3,17, 4,18)( 5,20, 6,19)( 7,11, 8,12)( 9,14,10,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $160=2^{5} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  160.207
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);