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Properties

Label 20T41
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\wr C_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 41);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr C_5$
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:  (1,6,10,13,17)(2,5,9,14,18)(3,7,11,16,20)(4,8,12,15,19), (1,2)(9,19)(10,20)(11,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$80$:  $C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$

Low degree siblings

10T14 x 3, 20T40 x 12, 20T41 x 5, 20T44 x 3, 20T46 x 3, 32T2133, 40T121 x 6, 40T122 x 6, 40T123 x 12, 40T141, 40T142 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  160.235
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3
Size 1 1 5 5 5 5 5 5 16 16 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1 5A-1 5A-2 5A1 5A2
5 P 1A 2A 2F 2B 2D 2C 2G 2E 1A 1A 1A 1A 2A 2A 2A 2A
Type
160.235.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
160.235.1b R 1 −1 1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
160.235.1c1 C 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2
160.235.1c2 C 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52
160.235.1c3 C 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1
160.235.1c4 C 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5
160.235.1d1 C 1 −1 1 −1 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2
160.235.1d2 C 1 −1 1 −1 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52
160.235.1d3 C 1 −1 1 −1 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1
160.235.1d4 C 1 −1 1 −1 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5
160.235.5a R 5 5 −3 −3 1 1 1 1 0 0 0 0 0 0 0 0
160.235.5b R 5 5 1 1 −3 1 −3 1 0 0 0 0 0 0 0 0
160.235.5c R 5 5 1 1 1 −3 1 −3 0 0 0 0 0 0 0 0
160.235.5d R 5 −5 −3 3 1 −1 −1 1 0 0 0 0 0 0 0 0
160.235.5e R 5 −5 1 −1 −3 −1 3 1 0 0 0 0 0 0 0 0
160.235.5f R 5 −5 1 −1 1 3 −1 −3 0 0 0 0 0 0 0 0

magma: CharacterTable(G);