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Properties

Label 20T26
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, 26);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15,20,10)(2,12,19,8)(3,14,18,6)(4,11,17,9)(5,13,16,7), (1,19,5,20,4,16,3,17,2,18)(6,11,9,13,7,15,10,12,8,14)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  $F_5$, 20T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

Low degree siblings

25T11

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 $2^{10}$ $5$ $2$ $10$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,14)( 7,13)( 8,12)( 9,11)(10,15)$
4A1 $4^{5}$ $25$ $4$ $15$ $( 1,15,20,10)( 2,12,19, 8)( 3,14,18, 6)( 4,11,17, 9)( 5,13,16, 7)$
4A-1 $4^{5}$ $25$ $4$ $15$ $( 1,10,20,15)( 2, 8,19,12)( 3, 6,18,14)( 4, 9,17,11)( 5, 7,16,13)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1, 4, 2, 5, 3)( 6, 7, 8, 9,10)(11,15,14,13,12)(16,18,20,17,19)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1, 5, 4, 3, 2)( 6, 9, 7,10, 8)(11,13,15,12,14)(16,17,18,19,20)$
5B $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 5, 4, 3, 2)(11,12,13,14,15)(16,18,20,17,19)$
5C1 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,13,15,12,14)$
5C-1 $5^{4}$ $4$ $5$ $16$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)$
5C2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)$
5C-2 $5^{3},1^{5}$ $4$ $5$ $12$ $( 1, 3, 5, 2, 4)( 6, 7, 8, 9,10)(16,20,19,18,17)$
10A1 $10^{2}$ $10$ $10$ $18$ $( 1,16, 4,18, 2,20, 5,17, 3,19)( 6,11, 7,15, 8,14, 9,13,10,12)$
10A3 $10^{2}$ $10$ $10$ $18$ $( 1,17, 2,16, 3,20, 4,19, 5,18)( 6,13, 8,11,10,14, 7,12, 9,15)$

Malle's constant $a(G)$:     $1/10$

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:  100.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A1 5A2 5B 5C1 5C-1 5C2 5C-2 10A1 10A3
Size 1 5 25 25 2 2 4 4 4 4 4 10 10
2 P 1A 1A 2A 2A 5A2 5A1 5C-2 5C1 5B 5C-1 5C2 5A1 5A2
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
100.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
100.10.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1
100.10.1c1 C 1 −1 −i i 1 1 1 1 1 1 1 −1 −1
100.10.1c2 C 1 −1 i −i 1 1 1 1 1 1 1 −1 −1
100.10.2a1 R 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
100.10.2a2 R 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
100.10.2b1 S 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
100.10.2b2 S 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
100.10.4a R 4 0 0 0 4 4 −1 −1 −1 −1 −1 0 0
100.10.4b1 C 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −2ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5 ζ5−2−ζ5−ζ52 1+ζ5+2ζ52 0 0
100.10.4b2 C 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−2+1+2ζ5 −2ζ5−2−1−2ζ5−ζ52 1+ζ5+2ζ52 ζ5−2−ζ5−ζ52 0 0
100.10.4b3 C 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 ζ5−2−ζ5−ζ52 1+ζ5+2ζ52 ζ5−2+1+2ζ5 −2ζ5−2−1−2ζ5−ζ52 0 0
100.10.4b4 C 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 1+ζ5+2ζ52 ζ5−2−ζ5−ζ52 −2ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5 0 0

magma: CharacterTable(G);