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Properties

Label 20T3
Degree $20$
Order $20$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(20, 3);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 10: $C_{10}$ x 3

Low degree siblings

There are no siblings 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, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 10, 10 $ $1$ $10$ $( 1, 3, 5, 8, 9,11,14,16,18,19)( 2, 4, 6, 7,10,12,13,15,17,20)$
$ 10, 10 $ $1$ $10$ $( 1, 4, 5, 7, 9,12,14,15,18,20)( 2, 3, 6, 8,10,11,13,16,17,19)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$
$ 10, 10 $ $1$ $10$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$
$ 10, 10 $ $1$ $10$ $( 1, 7,14,20, 5,12,18, 4, 9,15)( 2, 8,13,19, 6,11,17, 3,10,16)$
$ 10, 10 $ $1$ $10$ $( 1, 8,14,19, 5,11,18, 3, 9,16)( 2, 7,13,20, 6,12,17, 4,10,15)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$
$ 10, 10 $ $1$ $10$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
$ 10, 10 $ $1$ $10$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$
$ 10, 10 $ $1$ $10$ $( 1,15, 9, 4,18,12, 5,20,14, 7)( 2,16,10, 3,17,11, 6,19,13, 8)$
$ 10, 10 $ $1$ $10$ $( 1,16, 9, 3,18,11, 5,19,14, 8)( 2,15,10, 4,17,12, 6,20,13, 7)$
$ 10, 10 $ $1$ $10$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$
$ 10, 10 $ $1$ $10$ $( 1,19,18,16,14,11, 9, 8, 5, 3)( 2,20,17,15,13,12,10, 7, 6, 4)$
$ 10, 10 $ $1$ $10$ $( 1,20,18,15,14,12, 9, 7, 5, 4)( 2,19,17,16,13,11,10, 8, 6, 3)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $20=2^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  20.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1 5A-2 5A1 5A-2 5A-1 5A-1 5A1 5A2 5A-2 5A2 5A1 5A-1 5A2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 2C 2C 2B 2C 2B 2B 2B 2A 2C 2A 2A 2A
Type
20.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20.5.1b R 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
20.5.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
20.5.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
20.5.1e1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
20.5.1e2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
20.5.1e3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
20.5.1e4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
20.5.1f1 C 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−2 ζ52 ζ5−1 ζ5
20.5.1f2 C 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
20.5.1f3 C 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−1 ζ5 ζ52 ζ5−2
20.5.1f4 C 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
20.5.1g1 C 1 −1 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
20.5.1g2 C 1 −1 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
20.5.1g3 C 1 −1 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
20.5.1g4 C 1 −1 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
20.5.1h1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
20.5.1h2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
20.5.1h3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
20.5.1h4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52

magma: CharacterTable(G);