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Properties

Label 20T86
Degree $20$
Order $320$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5:C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(20, 86);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $86$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^5:C_{10}$
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:  (1,6,9,3,8,12,16,19,13,17)(2,5,10,4,7,11,15,20,14,18), (1,11)(2,12)(3,14)(4,13)(7,17)(8,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3
$20$:  20T3
$80$:  $C_2^4 : C_5$
$160$:  $C_2 \times (C_2^4 : C_5)$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_{10}$, $C_2 \times (C_2^4 : C_5)$ x 2

Low degree siblings

20T72 x 6, 20T74 x 36, 20T86 x 8, 40T202 x 3, 40T274 x 72, 40T275 x 36, 40T276 x 36, 40T277 x 18, 40T278 x 18, 40T292 x 9

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,20)(10,19)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,16)( 6,15)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3,14)( 4,13)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 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 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)(11,12)$
$ 10, 10 $ $16$ $10$ $( 1, 3, 5, 7, 9, 2, 4, 6, 8,10)(11,14,16,17,20,12,13,15,18,19)$
$ 10, 10 $ $16$ $10$ $( 1, 3, 5, 7, 9,12,13,15,18,19)( 2, 4, 6, 8,10,11,14,16,17,20)$
$ 5, 5, 5, 5 $ $16$ $5$ $( 1, 4, 5, 8, 9)( 2, 3, 6, 7,10)(11,13,16,18,20)(12,14,15,17,19)$
$ 10, 10 $ $16$ $10$ $( 1, 4, 5, 8, 9,11,13,16,18,20)( 2, 3, 6, 7,10,12,14,15,17,19)$
$ 5, 5, 5, 5 $ $16$ $5$ $( 1, 5, 9, 4, 8)( 2, 6,10, 3, 7)(11,16,20,13,18)(12,15,19,14,17)$
$ 10, 10 $ $16$ $10$ $( 1, 5, 9,13,18,11,16,20, 4, 8)( 2, 6,10,14,17,12,15,19, 3, 7)$
$ 10, 10 $ $16$ $10$ $( 1, 6, 9, 3, 8, 2, 5,10, 4, 7)(11,15,20,14,18,12,16,19,13,17)$
$ 10, 10 $ $16$ $10$ $( 1, 6, 9,14,18,12,16,19, 4, 7)( 2, 5,10,13,17,11,15,20, 3, 8)$
$ 10, 10 $ $16$ $10$ $( 1, 7, 4,10, 5, 2, 8, 3, 9, 6)(11,17,13,19,16,12,18,14,20,15)$
$ 10, 10 $ $16$ $10$ $( 1, 7, 4,10,16,12,18,14,20, 6)( 2, 8, 3, 9,15,11,17,13,19, 5)$
$ 5, 5, 5, 5 $ $16$ $5$ $( 1, 8, 4, 9, 5)( 2, 7, 3,10, 6)(11,18,13,20,16)(12,17,14,19,15)$
$ 10, 10 $ $16$ $10$ $( 1, 8, 4, 9,16,11,18,13,20, 5)( 2, 7, 3,10,15,12,17,14,19, 6)$
$ 5, 5, 5, 5 $ $16$ $5$ $( 1, 9, 8, 5, 4)( 2,10, 7, 6, 3)(11,20,18,16,13)(12,19,17,15,14)$
$ 10, 10 $ $16$ $10$ $( 1, 9,18,16,13,11,20, 8, 5, 4)( 2,10,17,15,14,12,19, 7, 6, 3)$
$ 10, 10 $ $16$ $10$ $( 1,10, 8, 6, 4, 2, 9, 7, 5, 3)(11,19,18,15,13,12,20,17,16,14)$
$ 10, 10 $ $16$ $10$ $( 1,10,18,15,13,12,20, 7, 5, 3)( 2, 9,17,16,14,11,19, 8, 6, 4)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $320=2^{6} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  320.1637
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);