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Properties

Label 20T10
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{20}$

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Show commands: Magma

magma: G := TransitiveGroup(20, 10);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T10, 40T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{10}$ $10$ $2$ $10$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,19)(14,20)(15,18)(16,17)$
2C $2^{9},1^{2}$ $10$ $2$ $9$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)(11,20)(12,19)(13,18)(14,17)(15,16)$
4A $4^{5}$ $2$ $4$ $15$ $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,17, 8,18)( 9,20,10,19)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1,10,18, 6,13)( 2, 9,17, 5,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1,18,13,10, 6)( 2,17,14, 9, 5)( 3,19,15,11, 7)( 4,20,16,12, 8)$
10A1 $10^{2}$ $2$ $10$ $18$ $( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 8,11,16,19, 4, 7,12,15,20)$
10A3 $10^{2}$ $2$ $10$ $18$ $( 1,14, 6,17,10, 2,13, 5,18, 9)( 3,16, 7,20,11, 4,15, 8,19,12)$
20A1 $20$ $2$ $20$ $19$ $( 1, 3, 5, 8,10,11,14,16,18,19, 2, 4, 6, 7, 9,12,13,15,17,20)$
20A3 $20$ $2$ $20$ $19$ $( 1,15, 9, 4,18,11, 5,20,13, 7, 2,16,10, 3,17,12, 6,19,14, 8)$
20A7 $20$ $2$ $20$ $19$ $( 1, 7,14,20, 6,11,17, 4,10,15, 2, 8,13,19, 5,12,18, 3, 9,16)$
20A9 $20$ $2$ $20$ $19$ $( 1,19,17,16,13,11, 9, 8, 6, 3, 2,20,18,15,14,12,10, 7, 5, 4)$

Malle's constant $a(G)$:     $1/9$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  40.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 10A1 10A3 20A1 20A3 20A7 20A9
Size 1 1 10 10 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A1 5A2 10A1 10A3 10A3 10A1
5 P 1A 2A 2B 2C 4A 1A 1A 2A 2A 4A 4A 4A 4A
Type
40.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.6.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1
40.6.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.6.1d R 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
40.6.2a R 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.6.2b1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
40.6.2b2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
40.6.2c1 R 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
40.6.2c2 R 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
40.6.2d1 R 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 −ζ20−3−ζ203 ζ20−1+ζ20 −ζ20−1−ζ20 ζ20−3+ζ203
40.6.2d2 R 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 ζ20−3+ζ203 −ζ20−1−ζ20 ζ20−1+ζ20 −ζ20−3−ζ203
40.6.2d3 R 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ20−1−ζ20 −ζ20−3−ζ203 ζ20−3+ζ203 ζ20−1+ζ20
40.6.2d4 R 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 ζ20−1+ζ20 ζ20−3+ζ203 −ζ20−3−ζ203 −ζ20−1−ζ20

magma: CharacterTable(G);