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Properties

Label 20T27
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 27);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:F_5$
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:  (1,12,18,7)(2,11,17,8)(3,5,15,14)(4,6,16,13)(9,20,10,19), (3,8,12,16,19)(4,7,11,15,20)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

Low degree siblings

10T10 x 2, 20T27, 25T10

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 $2^{10}$ $25$ $2$ $10$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$
4A1 $4^{5}$ $25$ $4$ $15$ $( 1,12,18, 7)( 2,11,17, 8)( 3, 5,15,14)( 4, 6,16,13)( 9,20,10,19)$
4A-1 $4^{5}$ $25$ $4$ $15$ $( 1, 7,18,12)( 2, 8,17,11)( 3,14,15, 5)( 4,13,16, 6)( 9,19,10,20)$
5A $5^{4}$ $4$ $5$ $16$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,12,19, 8,16)( 4,11,20, 7,15)$
5B $5^{4}$ $4$ $5$ $16$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3, 8,12,16,19)( 4, 7,11,15,20)$
5C1 $5^{4}$ $4$ $5$ $16$ $( 1,17,14,10, 6)( 2,18,13, 9, 5)( 3,19,16,12, 8)( 4,20,15,11, 7)$
5C2 $5^{2},1^{10}$ $4$ $5$ $8$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)$
5D1 $5^{2},1^{10}$ $4$ $5$ $8$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$
5D2 $5^{4}$ $4$ $5$ $16$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  100.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A 5B 5C1 5C2 5D1 5D2
Size 1 25 25 25 4 4 4 4 4 4
2 P 1A 1A 2A 2A 5D1 5A 5D2 5C2 5C1 5B
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A
Type
100.12.1a R 1 1 1 1 1 1 1 1 1 1
100.12.1b R 1 1 −1 −1 1 1 1 1 1 1
100.12.1c1 C 1 −1 −i i 1 1 1 1 1 1
100.12.1c2 C 1 −1 i −i 1 1 1 1 1 1
100.12.4a R 4 0 0 0 −1 4 −1 −1 −1 −1
100.12.4b R 4 0 0 0 4 −1 −1 −1 −1 −1
100.12.4c1 R 4 0 0 0 −1 −1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52
100.12.4c2 R 4 0 0 0 −1 −1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52
100.12.4d1 R 4 0 0 0 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52
100.12.4d2 R 4 0 0 0 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5

magma: CharacterTable(G);