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Properties

Label 20T48
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(20, 48);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\wr C_2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,9,19)(2,8,10,20)(3,6,4,5)(11,13,16,17)(12,14,15,18), (1,2)(3,19,16,12,8)(4,20,15,11,7)(5,17)(6,18)(9,14)(10,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: None

Degree 10: $D_5^2 : C_2$

Low degree siblings

10T19, 10T21 x 2, 20T48, 20T50 x 2, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 7,19)( 8,20)(11,16)(12,15)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $25$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7,19)( 8,20)( 9,14)(10,13)(11,16)(12,15)$
$ 5, 5, 2, 2, 2, 2, 2 $ $20$ $10$ $( 1, 2)( 3, 8,12,16,19)( 4, 7,11,15,20)( 5,17)( 6,18)( 9,14)(10,13)$
$ 5, 5, 2, 2, 2, 2, 2 $ $20$ $10$ $( 1, 2)( 3,12,19, 8,16)( 4,11,20, 7,15)( 5,17)( 6,18)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$
$ 4, 4, 4, 4, 4 $ $50$ $4$ $( 1, 3, 2, 4)( 5, 7,17,19)( 6, 8,18,20)( 9,11,14,16)(10,12,13,15)$
$ 10, 10 $ $20$ $10$ $( 1, 3, 6, 8,10,12,14,16,17,19)( 2, 4, 5, 7, 9,11,13,15,18,20)$
$ 10, 10 $ $20$ $10$ $( 1, 3,10,12,17,19, 6, 8,14,16)( 2, 4, 9,11,18,20, 5, 7,13,15)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3, 8,12,16,19)( 4, 7,11,15,20)$
$ 5, 5, 5, 5 $ $8$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,12,19, 8,16)( 4,11,20, 7,15)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,12,19, 8,16)( 4,11,20, 7,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  200.43
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 5B1 5B2 5C 10A1 10A3 10B1 10B3
Size 1 10 10 25 50 4 4 4 4 8 20 20 20 20
2 P 1A 1A 1A 1A 2C 5A1 5B2 5A2 5B1 5C 5A1 5B1 5B2 5A2
5 P 1A 2A 2B 2C 4A 1A 1A 1A 1A 1A 2A 2B 2B 2A
Type
200.43.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.43.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
200.43.1c R 1 −1 1 1 −1 1 1 1 1 1 −1 −1 1 1
200.43.1d R 1 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1
200.43.2a R 2 0 0 −2 0 2 2 2 2 2 0 0 0 0
200.43.4a1 R 4 0 2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 ζ5−2+ζ52 ζ5−1+ζ5
200.43.4a2 R 4 0 2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 ζ5−1+ζ5 ζ5−2+ζ52
200.43.4b1 R 4 2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−1+ζ5 ζ5−2+ζ52 0 0
200.43.4b2 R 4 2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 ζ5−2+ζ52 ζ5−1+ζ5 0 0
200.43.4c1 R 4 −2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
200.43.4c2 R 4 −2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
200.43.4d1 R 4 0 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
200.43.4d2 R 4 0 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
200.43.8a R 8 0 0 0 0 −2 −2 −2 −2 3 0 0 0 0

magma: CharacterTable(G);