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Properties

Label 20T7
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(20, 7);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:D_4$
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:  (1,11,2,12)(3,9,4,10)(5,8,6,7)(13,19,14,20)(15,17,16,18), (1,8,13,20,6,12,18,4,10,16)(2,7,14,19,5,11,17,3,9,15)
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

20T11, 40T11

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
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 $2^{10}$ $2$ $2$ $10$ $( 1,12)( 2,11)( 3,14)( 4,13)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
2C $2^{9},1^{2}$ $10$ $2$ $9$ $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,13)(10,14)$
4A $4^{5}$ $10$ $4$ $15$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,20,14,19)(15,18,16,17)$
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 $10^{2}$ $2$ $10$ $18$ $( 1,14, 6,17,10, 2,13, 5,18, 9)( 3,16, 7,20,11, 4,15, 8,19,12)$
10B1 $10^{2}$ $2$ $10$ $18$ $( 1,15,10, 3,18,11, 6,19,13, 7)( 2,16, 9, 4,17,12, 5,20,14, 8)$
10B-1 $10^{2}$ $2$ $10$ $18$ $( 1,19,18,15,13,11,10, 7, 6, 3)( 2,20,17,16,14,12, 9, 8, 5, 4)$
10B3 $10^{2}$ $2$ $10$ $18$ $( 1, 7,13,19, 6,11,18, 3,10,15)( 2, 8,14,20, 5,12,17, 4, 9,16)$
10B-3 $10^{2}$ $2$ $10$ $18$ $( 1, 3, 6, 7,10,11,13,15,18,19)( 2, 4, 5, 8, 9,12,14,16,17,20)$

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:  40.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 10A1 10A3 10B1 10B-1 10B3 10B-3
Size 1 1 2 10 10 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A1 5A2 5A1 5A2 5A1 5A2
5 P 1A 2A 2B 2C 4A 1A 1A 2A 2A 2B 2B 2B 2B
Type
40.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.8.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
40.8.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.8.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.8.2a R 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.8.2b1 R 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
40.8.2b2 R 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
40.8.2c1 R 2 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
40.8.2c2 R 2 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
40.8.2d1 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52
40.8.2d2 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52
40.8.2d3 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52
40.8.2d4 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52

magma: CharacterTable(G);