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Properties

Label 20T43
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:D_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 43);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:D_5$
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:  (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,9,17,19)(8,10,18,20), (1,13,11,3)(2,14,12,4)(5,9)(6,10)(7,18)(8,17)(15,19)(16,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $D_5$, $(C_2^4 : C_5) : C_2$, $(C_2^4 : C_5) : C_2$

Low degree siblings

10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 20T39, 20T43 x 2, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

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 $ $5$ $2$ $( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,15)( 6,16)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12)(13,19)(14,20)(15,18)(16,17)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8,15,18)( 6, 7,16,17)(11,12)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 3)( 2, 4)( 5, 9,15,19)( 6,10,16,20)( 7,18)( 8,17)(11,13)(12,14)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 3,11,13)( 2, 4,12,14)( 5, 9)( 6,10)( 7,18)( 8,17)(15,19)(16,20)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10)(11,14,16,18,19)(12,13,15,17,20)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1, 6, 9, 4, 8)( 2, 5,10, 3, 7)(11,16,19,14,18)(12,15,20,13,17)$

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:  160.234
magma: IdentifyGroup(G);
 
Character table:    
      2  5  5  5  5  3  3  3  3  .  .
      5  1  .  .  .  .  .  .  .  1  1

        1a 2a 2b 2c 2d 4a 4b 4c 5a 5b
     2P 1a 1a 1a 1a 1a 2c 2b 2a 5b 5a
     3P 1a 2a 2b 2c 2d 4a 4b 4c 5b 5a
     5P 1a 2a 2b 2c 2d 4a 4b 4c 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1 -1 -1 -1 -1  1  1
X.3      2  2  2  2  .  .  .  .  A *A
X.4      2  2  2  2  .  .  .  . *A  A
X.5      5 -3  1  1 -1  1  1 -1  .  .
X.6      5 -3  1  1  1 -1 -1  1  .  .
X.7      5  1 -3  1 -1  1 -1  1  .  .
X.8      5  1 -3  1  1 -1  1 -1  .  .
X.9      5  1  1 -3 -1 -1  1  1  .  .
X.10     5  1  1 -3  1  1 -1 -1  .  .

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5

magma: CharacterTable(G);